Proving Angle Addition/Subtraction Formulas - Precalculus

Use the angle addition formulas to simplify $\sin\left(x+\dfrac{\pi}{4}\right)+\sin\left(x-\dfrac{\pi}{4}\right)$ . What is the simplified expression?
Answer: $2\sin(x)\cos\left(\dfrac{\pi}{4}\right)$ This question tests understanding of the angle addition and subtraction formulas for sine. The sine addition formula is sin(A + B) = sin(A)cos(B) + cos(A)sin(B), which shows that sine of a sum is NOT simply sin(A) + sin(B) but requires cross terms involving both sine and cosine of each angle. Expanding sin(x + π/4) + sin(x - π/4) gives [sin(x)cos(π/4) + cos(x)sin(π/4)] + [sin(x)cos(π/4) - cos(x)sin(π/4)] = 2 sin(x) cos(π/4), since the cos(x) terms cancel. Choice A is correct because it properly substitutes the angle values and simplifies accurately. Choice C incorrectly claims sin(A + B) = sin(A) + sin(B), missing the essential cross terms that make the actual formula sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Key to angle addition formulas: memorize that both sine and cosine formulas involve four terms (two cross products), with sine having + for addition and - for subtraction, while cosine has - for addition and + for subtraction (opposite signs). To use these formulas for exact values, break unfamiliar angles into sums or differences of standard angles (like 75° = 45° + 30° or 15° = 45° - 30°), then apply the formula with the known exact trig values.
Which formula correctly gives the cosine subtraction formula for $\cos(A-B)$ (be careful about the sign pattern)?
Answer: $\cos(A-B)=\cos(A)\cos(B)+\sin(A)\sin(B)$ This question tests understanding of the angle addition and subtraction formulas for cosine. The cosine addition formula is cos(A + B) = cos(A)cos(B) - sin(A)sin(B), and the subtraction formula is cos(A - B) = cos(A)cos(B) + sin(A)sin(B), with the key difference being the sign between the two product terms (minus for addition, plus for subtraction). Using cos(A - B) = cos(A)cos(B) + sin(A)sin(B), we see that the subtraction formula requires the plus sign to account for the difference in angles. Choice C is correct because it correctly states the formula with proper signs. Choice A reverses the subtraction formula, using cos(A)cos(B) - sin(A)sin(B) for cos(A - B) when actually that's the addition formula; the subtraction formula is cos(A)cos(B) + sin(A)sin(B). Key to angle addition formulas: memorize that both sine and cosine formulas involve four terms (two cross products), with sine having + for addition and - for subtraction, while cosine has - for addition and + for subtraction (opposite signs). Remember the pattern: sin formulas have sin·cos + cos·sin (same functions in each term), while cos formulas have cos·cos ∓ sin·sin (same functions in each term but different from numerator function), and the signs are opposite between sin and cos formulas.
Use the cosine angle subtraction formula to find the exact value of $\cos(15^\circ)$ by rewriting it as $\cos(45^\circ-30^\circ)$ . What is the exact value?
Answer: $\dfrac{\sqrt{6}+\sqrt{2}}{4}$ This question tests understanding of the angle addition and subtraction formulas for cosine. The cosine addition formula is cos(A + B) = cos(A)cos(B) - sin(A)sin(B), and the subtraction formula is cos(A - B) = cos(A)cos(B) + sin(A)sin(B), with the key difference being the sign between the two product terms (minus for addition, plus for subtraction). Using cos(A - B) = cos(A)cos(B) + sin(A)sin(B) with A = 45°, B = 30°, we calculate cos(45°)cos(30°) + sin(45°)sin(30°) = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4. Choice B is correct because it properly substitutes the angle values and simplifies accurately. Choice A has the wrong sign between the terms, using minus when the formula for cosine subtraction requires plus. To use these formulas for exact values, break unfamiliar angles into sums or differences of standard angles (like 75° = 45° + 30° or 15° = 45° - 30°), then apply the formula with the known exact trig values. Key to angle addition formulas: memorize that both sine and cosine formulas involve four terms (two cross products), with sine having + for addition and - for subtraction, while cosine has - for addition and + for subtraction (opposite signs).
Using the sine angle addition formula, find the exact value of $\sin(75^\circ)$ by rewriting it as $\sin(45^\circ+30^\circ)$ . What is the exact value?
Answer: $\dfrac{\sqrt{6}+\sqrt{2}}{4}$ This question tests understanding of the angle addition and subtraction formulas for sine. The sine addition formula is sin(A + B) = sin(A)cos(B) + cos(A)sin(B), which shows that sine of a sum is NOT simply sin(A) + sin(B) but requires cross terms involving both sine and cosine of each angle. To find sin(75°), we recognize 75° = 45° + 30°, so sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°) = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4. Choice B is correct because it properly substitutes the angle values and simplifies accurately. Choice A has the wrong sign between the terms, using minus when the formula for sine addition requires plus. To use these formulas for exact values, break unfamiliar angles into sums or differences of standard angles (like 75° = 45° + 30° or 15° = 45° - 30°), then apply the formula with the known exact trig values. Common error: students try sin(A + B) = sin(A) + sin(B), but you can quickly verify this is wrong by trying A = B = 45°: sin(90°) = 1 but sin(45°) + sin(45°) = √2/2 + √2/2 = √2 ≠ 1.
Use angle addition formulas to simplify the expression $\sin\left(x+\dfrac{\pi}{4}\right)+\sin\left(x-\dfrac{\pi}{4}\right)$ .
Answer: $\sqrt{2}\sin(x)$ This question tests understanding of the angle addition and subtraction formulas for sine. The sine addition formula is sin(A + B) = sin(A)cos(B) + cos(A)sin(B), which shows that sine of a sum is NOT simply sin(A) + sin(B) but requires cross terms involving both sine and cosine of each angle. Expanding sin(x + π/4) + sin(x - π/4) gives [sin(x)cos(π/4) + cos(x)sin(π/4)] + [sin(x)cos(π/4) - cos(x)sin(π/4)] = 2 sin(x) cos(π/4) = 2 sin(x) (√2/2) = √2 sin(x). Choice A is correct because it properly substitutes the angle values and simplifies accurately. Choice B confuses the sine formula with the cosine formula, using a cosine result when sine requires sin(A)cos(B) + cos(A)sin(B) for addition and subtraction accordingly. Key to angle addition formulas: memorize that both sine and cosine formulas involve four terms (two cross products), with sine having + for addition and - for subtraction, while cosine has - for addition and + for subtraction (opposite signs). To use these formulas for exact values, break unfamiliar angles into sums or differences of standard angles, then apply the formula with the known exact trig values.
Which formula correctly gives the sine angle addition identity $\sin(A+B)$ ?
Answer: $\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$ This question tests understanding of the angle addition formula for sine. The sine addition formula is sin(A + B) = sin(A)cos(B) + cos(A)sin(B), which shows that sine of a sum is NOT simply sin(A) + sin(B) but requires cross terms involving both sine and cosine of each angle. The formula specifically requires the cross products: sin(A) multiplied by cos(B), plus cos(A) multiplied by sin(B), with a positive sign between them. Choice C is correct because it correctly states the formula with proper signs and cross terms: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Choice B incorrectly claims sin(A + B) = sin(A) + sin(B), missing the essential cross terms that make the actual formula sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Common error: students try sin(A + B) = sin(A) + sin(B), but you can quickly verify this is wrong by trying A = B = 45°: sin(90°) = 1 but sin(45°) + sin(45°) = √2/2 + √2/2 = √2 ≠ 1.
Which of the following correctly represents the derivation step when proving the tangent subtraction formula $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ from the sine and cosine subtraction formulas?
Answer: $\tan(A - B) = \frac{\sin(A - B)}{\cos(A - B)} = \frac{\sin A \cos B - \cos A \sin B}{\cos A \cos B + \sin A \sin B}$ The correct derivation uses $\tan(A - B) = \frac{\sin(A - B)}{\cos(A - B)}$ , then applies the sine and cosine subtraction formulas: $\sin(A - B) = \sin A \cos B - \cos A \sin B$ and $\cos(A - B) = \cos A \cos B + \sin A \sin B$ . Option B incorrectly uses the sine addition formula in the numerator. Option C incorrectly inverts the tangent definition as $\frac{\cos}{\sin}$ . Option D uses the wrong cosine formula (subtraction instead of the correct form for $\cos(A-B)$ ).
Which formula correctly gives the cosine subtraction identity $\cos(A-B)$ in terms of $\sin$ and $\cos$ ?
Answer: $\cos(A-B)=\cos(A)\cos(B)+\sin(A)\sin(B)$ This question tests understanding of the angle subtraction formula for cosine. The cosine addition formula is cos(A + B) = cos(A)cos(B) - sin(A)sin(B), and the subtraction formula is cos(A - B) = cos(A)cos(B) + sin(A)sin(B), with the key difference being the sign between the two product terms (minus for addition, plus for subtraction). For cosine subtraction specifically, we have cos(A - B) = cos(A)cos(B) + sin(A)sin(B), where the plus sign between the terms is crucial. Choice B is correct because it correctly states the formula with proper signs: cos(A - B) = cos(A)cos(B) + sin(A)sin(B). Choice A incorrectly has the wrong sign between the terms, using cos(A)cos(B) - sin(A)sin(B) which is actually the cosine addition formula for cos(A + B), not the subtraction formula. Remember the pattern: sin formulas have sin·cos + cos·sin (same functions in each term), while cos formulas have cos·cos ∓ sin·sin (same functions in each term but different from numerator function), and the signs are opposite between sin and cos formulas.
Using the tangent angle subtraction formula, find the exact value of $\tan(15^\circ)$ by writing $15^\circ = 45^\circ - 30^\circ$ .
Answer: $2-\sqrt{3}$ This question tests understanding of the angle subtraction formula for tangent. The tangent subtraction formula is tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B)), which shows that tangent of a difference involves both a numerator (difference of tangents) and a denominator (1 plus their product). Applying tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B)) with tan(45°) = 1 and tan(30°) = 1/√3 = √3/3, we get (1 - √3/3)/(1 + 1·√3/3) = (3/3 - √3/3)/(3/3 + √3/3) = (3 - √3)/(3 + √3). To simplify, multiply numerator and denominator by (3 - √3): [(3 - √3)(3 - √3)]/[(3 + √3)(3 - √3)] = (9 - 6√3 + 3)/(9 - 3) = (12 - 6√3)/6 = 2 - √3. Choice B is correct because it properly substitutes the angle values and simplifies accurately to 2 - √3. Choice A incorrectly gives just √3, which is actually tan(60°), not tan(15°). To use these formulas for exact values, break unfamiliar angles into sums or differences of standard angles (like 75° = 45° + 30° or 15° = 45° - 30°), then apply the formula with the known exact trig values.
Using the tangent angle addition formula, what is $\tan(A+B)$ in terms of $\tan(A)$ and $\tan(B)$ ?
Answer: $\tan(A+B)=\dfrac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)}$ This question tests understanding of the angle addition and subtraction formulas for tangent. The tangent addition formula is tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)), which shows that tangent of a sum involves both a numerator (sum of tangents) and a denominator (1 minus their product). Applying tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)) directly gives the required expression. Choice A is correct because it correctly states the formula with proper signs. Choice B has the wrong sign between the terms, using plus in the denominator when the formula for tangent addition requires minus. For tangent, remember the formulas have fractions: tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)), with the sign in the denominator opposite to the numerator (minus for addition, plus for subtraction). Common error: students try tan(A + B) = tan(A) + tan(B), but you can quickly verify this is wrong by trying A = B = 45°: tan(90°) is undefined but tan(45°) + tan(45°) = 1 + 1 = 2.
A calculus student needs to prove that $\frac{d}{dx}[\sin(x + h)] = \cos(x + h)$ and plans to use the sine addition formula as an intermediate step. Which expression correctly represents the application of $\sin(A + B) = \sin A \cos B + \cos A \sin B$ to $\sin(x + h)$ ?
Answer: $\sin(x + h) = \sin x \cos h + \cos x \sin h$ , which allows separation of terms involving $x$ and $h$ for differentiation When working with trigonometric identities and derivatives, the sine addition formula is a fundamental tool that breaks down complex expressions into manageable parts. The formula $\sin(A + B) = \sin A \cos B + \cos A \sin B$ allows you to expand any sine of a sum into terms that can be differentiated separately. For $\sin(x + h)$ , you need to identify $A = x$ and $B = h$ , then apply the formula directly. This gives you $\sin(x + h) = \sin x \cos h + \cos x \sin h$ . This expansion is crucial for differentiation because it separates the variable $x$ (which you're differentiating with respect to) from the parameter $h$ , making it possible to apply derivative rules to each term individually. Choice A reverses the order of terms in the addition formula, writing $\sin h \cos x + \cos h \sin x$ instead of the correct $\sin x \cos h + \cos x \sin h$ . While mathematically equivalent due to commutativity, this doesn't match the standard application of the sine addition formula. Choice B incorrectly uses $\sin x \sin h + \cos x \cos h$ , which is actually the cosine addition formula $\cos(x - h)$ , not the sine addition formula. Choice C gives $\cos x \cos h - \sin x \sin h$ , which is the cosine addition formula for $\cos(x + h)$ , completely wrong for expanding $\sin(x + h)$ . Remember: always match the trigonometric function in your expansion to the function you're working with. Sine addition formulas expand sines, cosine addition formulas expand cosines.
Given that $\cos(75°) = \frac{\sqrt{6} - \sqrt{2}}{4}$ , which angle addition or subtraction formula application most directly verifies this result?
Answer: $\cos(75°) = \cos(45° + 30°) = \cos(45°)\cos(30°) - \sin(45°)\sin(30°)$ When you encounter problems asking you to verify trigonometric values using angle formulas, look for angle combinations that break down into familiar reference angles (30°, 45°, 60°) whose exact trigonometric values you know by heart. Let's verify the given result by testing option C: $\cos(75°) = \cos(45° + 30°)$ . Using the cosine addition formula $\cos(A + B) = \cos A \cos B - \sin A \sin B$ : $\cos(75°) = \cos(45°)\cos(30°) - \sin(45°)\sin(30°)$ $= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ $= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}$ This matches the given result perfectly. Option A uses $\cos(90° - 15°)$ , but this becomes $\sin(15°)$ by the cofunction identity, not the cosine addition formula shown. Option B uses $\cos(120° - 45°)$ with angles whose values we know, but $\cos(120°) = -\frac{1}{2}$ and $\sin(120°) = \frac{\sqrt{3}}{2}$ , which won't yield the correct result when calculated. Option D uses $\cos(135° - 60°)$ , but $\cos(135°) = -\frac{\sqrt{2}}{2}$ , and the negative value will prevent getting the positive result we need. Study tip: When verifying trigonometric identities, always choose angle combinations that use 30°, 45°, and 60° since you can calculate their exact values without a calculator. Avoid angles that introduce negative values unless the final result should be negative.
Which formula correctly gives $\sin(A+B)$ (and is not the incorrect idea $\sin(A)+\sin(B)$ )?
Answer: $\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$ This question tests understanding of the angle addition and subtraction formulas for sine. The sine addition formula is sin(A + B) = sin(A)cos(B) + cos(A)sin(B), which shows that sine of a sum is NOT simply sin(A) + sin(B) but requires cross terms involving both sine and cosine of each angle. If sin(A + B) simply equaled sin(A) + sin(B), then sin(30° + 60°) = sin(90°) = 1 would equal sin(30°) + sin(60°) = 1/2 + √3/2 ≈ 1.37, which is false, demonstrating that the formula requires the cross terms sin(A)cos(B) + cos(A)sin(B) instead. Choice A is correct because it correctly states the formula with proper signs. Choice D incorrectly claims sin(A + B) = sin(A) + sin(B), missing the essential cross terms that make the actual formula sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Common error: students try sin(A + B) = sin(A) + sin(B), but you can quickly verify this is wrong by trying A = B = 45°: sin(90°) = 1 but sin(45°) + sin(45°) = √2/2 + √2/2 = √2 ≠ 1. Remember the pattern: sin formulas have sin·cos + cos·sin (same functions in each term), while cos formulas have cos·cos ∓ sin·sin (same functions in each term but different from numerator function), and the signs are opposite between sin and cos formulas.
A proof of the cosine addition formula $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ uses the unit circle and the distance formula between points $P(\cos(\alpha + \beta), \sin(\alpha + \beta))$ and $Q(1, 0)$ . What is the key insight that completes this proof?
Answer: The distance $PQ$ equals the distance between points $R(\cos \alpha, \sin \alpha)$ and $S(\cos(-\beta), \sin(-\beta))$ on the unit circle The key insight is that rotating the unit circle preserves distances. The distance from $P(\cos(\alpha + \beta), \sin(\alpha + \beta))$ to $Q(1, 0)$ equals the distance from $R(\cos \alpha, \sin \alpha)$ to $S(\cos(-\beta), \sin(-\beta))$ because both represent the same angular separation on the unit circle. Setting these distances equal using the distance formula and simplifying yields the cosine addition formula. Option B incorrectly suggests rotation changes the relationship. Option C misunderstands that we need equal distances, not arc length. Option D uses the wrong point $S$ , which would lead to incorrect algebra.
Using the sine angle addition formula, find the exact value of $\sin(75^\circ)$ by writing $75^\circ = 45^\circ + 30^\circ$ .
Answer: $\dfrac{\sqrt{6}+\sqrt{2}}{4}$ This question tests understanding of the angle addition formula for sine. The sine addition formula is sin(A + B) = sin(A)cos(B) + cos(A)sin(B), which shows that sine of a sum is NOT simply sin(A) + sin(B) but requires cross terms involving both sine and cosine of each angle. To find sin(75°), we recognize 75° = 45° + 30°, so sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°) = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4. Choice B is correct because it properly substitutes the angle values and simplifies accurately to (√6 + √2)/4. Choice A incorrectly has (√6 - √2)/4, which would result from using the wrong sign between the terms or confusing this with a cosine formula. Remember the pattern: sin formulas have sin·cos + cos·sin (same functions in each term), while cos formulas have cos·cos ∓ sin·sin (same functions in each term but different from numerator function), and the signs are opposite between sin and cos formulas.
Using the cosine angle subtraction formula, find the exact value of $\cos(15^\circ)$ by writing $15^\circ = 45^\circ - 30^\circ$ .
Answer: $\dfrac{\sqrt{6}+\sqrt{2}}{4}$ This question tests understanding of the angle subtraction formula for cosine. The cosine addition formula is cos(A + B) = cos(A)cos(B) - sin(A)sin(B), and the subtraction formula is cos(A - B) = cos(A)cos(B) + sin(A)sin(B), with the key difference being the sign between the two product terms (minus for addition, plus for subtraction). Using cos(A - B) = cos(A)cos(B) + sin(A)sin(B) with A = 45°, B = 30°, we calculate cos(45°)cos(30°) + sin(45°)sin(30°) = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4. Choice A is correct because it correctly applies the cosine subtraction formula with the proper positive sign between terms. Choice B incorrectly has (√6 - √2)/4, which would result from using the cosine addition formula instead of subtraction, mixing up the sign convention. Key to angle addition formulas: memorize that both sine and cosine formulas involve four terms (two cross products), with sine having + for addition and - for subtraction, while cosine has - for addition and + for subtraction (opposite signs).
Which statement best explains why $\sin(A+B)\ne \sin(A)+\sin(B)$ in general, referring to the sine angle addition formula?
Answer: Because $\sin(A+B)$ must include the cross terms $\sin(A)\cos(B)$ and $\cos(A)\sin(B)$ , not just a sum of sines. This question tests understanding of why the angle addition formula for sine requires cross terms. The sine addition formula is sin(A + B) = sin(A)cos(B) + cos(A)sin(B), which shows that sine of a sum is NOT simply sin(A) + sin(B) but requires cross terms involving both sine and cosine of each angle. If sin(A + B) simply equaled sin(A) + sin(B), then sin(30° + 60°) = sin(90°) = 1 would equal sin(30°) + sin(60°) = 1/2 + √3/2 ≈ 1.37, which is false, demonstrating that the formula requires the cross terms sin(A)cos(B) + cos(A)sin(B) instead. Choice A is correct because it explains why cross terms are necessary: the sine of a sum must include the cross terms sin(A)cos(B) and cos(A)sin(B), not just a sum of sines. Choice D incorrectly focuses on the range of values rather than the structural reason why the formula needs cross terms - the issue isn't about bounds but about the fundamental trigonometric relationship. Common error: students try sin(A + B) = sin(A) + sin(B), but you can quickly verify this is wrong by trying A = B = 45°: sin(90°) = 1 but sin(45°) + sin(45°) = √2/2 + √2/2 = √2 ≠ 1.
Use the tangent angle addition formula to write $\tan(A+B)$ in terms of $\tan(A)$ and $\tan(B)$ .
Answer: $\tan(A+B)=\dfrac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)}$ This question tests understanding of the angle addition formula for tangent. The tangent addition formula is tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)), which shows that tangent of a sum involves both a numerator (sum of tangents) and a denominator (1 minus their product). The formula has a fraction structure where the numerator contains the sum tan(A) + tan(B), and the denominator contains 1 - tan(A)tan(B), with a minus sign being crucial for the addition formula. Choice A is correct because it correctly states the formula with proper signs: tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)). Choice B incorrectly has the wrong sign in the denominator, using 1 + tan(A)tan(B) when the formula for tan(A + B) requires the denominator to be 1 - tan(A)tan(B). For tangent, remember the formulas have fractions: tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)), with the sign in the denominator opposite to the numerator (minus for addition, plus for subtraction).
Which formula correctly gives $\sin(A+B)$ (sine angle addition formula)?
Answer: $\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$ This question tests understanding of the angle addition and subtraction formulas for sine. The sine addition formula is sin(A + B) = sin(A)cos(B) + cos(A)sin(B), which shows that sine of a sum is NOT simply sin(A) + sin(B) but requires cross terms involving both sine and cosine of each angle. The formula sin(A + B) = sin(A)cos(B) + cos(A)sin(B) directly matches the required addition formula. Choice B is correct because it correctly states the formula with proper signs. Choice A incorrectly claims sin(A + B) = sin(A) + sin(B), missing the essential cross terms that make the actual formula sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Key to angle addition formulas: memorize that both sine and cosine formulas involve four terms (two cross products), with sine having + for addition and - for subtraction, while cosine has - for addition and + for subtraction (opposite signs). Common error: students try sin(A + B) = sin(A) + sin(B), but you can quickly verify this is wrong by trying A = B = 45°: sin(90°) = 1 but sin(45°) + sin(45°) = √2/2 + √2/2 = √2 ≠ 1.
Use the sine angle subtraction formula to find the exact value of $\sin(15^\circ)$ by writing it as $\sin(45^\circ-30^\circ)$ .
Answer: $\dfrac{\sqrt{6}-\sqrt{2}}{4}$ This question tests understanding of the angle addition and subtraction formulas for sine. The sine addition formula is $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$ , which shows that sine of a sum is NOT simply $\sin(A) + \sin(B)$ but requires cross terms involving both sine and cosine of each angle. To find $\sin(15^\circ)$ , we recognize $15^\circ = 45^\circ - 30^\circ$ , so $\sin(45^\circ - 30^\circ) = \sin(45^\circ)\cos(30^\circ) - \cos(45^\circ)\sin(30^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}$ . Choice B is correct because it properly substitutes the angle values and simplifies accurately. Choice A has the wrong sign between the terms, using addition when the formula for sine subtraction requires subtraction. To use these formulas for exact values, break unfamiliar angles into sums or differences of standard angles (like $75^\circ = 45^\circ + 30^\circ$ or $15^\circ = 45^\circ - 30^\circ$ ), then apply the formula with the known exact trig values. Key to angle addition formulas: memorize that both sine and cosine formulas involve four terms (two cross products), with sine having + for addition and - for subtraction, while cosine has - for addition and + for subtraction (opposite signs).

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