Proving Angle Addition/Subtraction Formulas - Geometry

1. State the formula for $\sin(\alpha-\beta)$.

   $\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$. Standard angle subtraction formula for sine function.

2. State the cofunction identity for tangent: what is $\tan\left(\frac{\pi}{2}-\theta\right)$?

   $\tan\left(\frac{\pi}{2}-\theta\right)=\frac{1}{\tan\theta}$. Tangent cofunction identity equivalent to $\cot\theta$.

3. State the cofunction identity for sine: what is $\sin\left(\frac{\pi}{2}-\theta\right)$?

   $\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta$. Sine and cosine are cofunctions differing by $\frac{\pi}{2}$.

4. Identify the condition that makes $\tan(\alpha+\beta)$ undefined in $\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$.

   $1-\tan\alpha\tan\beta=0$. When denominator equals zero, tangent is undefined.

5. State the double-angle identity for sine: what is $\sin(2\theta)$?

   $\sin(2\theta)=2\sin\theta\cos\theta$. Double-angle formula derived from sine addition formula.

6. State the formula for $\tan(\alpha+\beta)$.

   $\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$. Standard angle addition formula for tangent function.

7. Identify the correct identity for $\cos(\alpha+\beta)$ among sign choices before $\sin\alpha\sin\beta$.

   $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. The correct sign before $\sin\alpha\sin\beta$ in cosine addition is negative.

8. State one standard form of the double-angle identity for cosine: what is $\cos(2\theta)$?

   $\cos(2\theta)=\cos^2\theta-\sin^2\theta$. Double-angle formula derived from cosine addition formula.

9. Identify the expression equivalent to $\sin(\beta-\alpha)$ in terms of $\sin(\alpha-\beta)$.

   $\sin(\beta-\alpha)=-\sin(\alpha-\beta)$. Sine is an odd function, so $\sin(\beta-\alpha) = -\sin(\alpha-\beta)$.

10. Identify the condition that makes $\tan(\alpha-\beta)$ undefined in $\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$.

    $1+\tan\alpha\tan\beta=0$. When denominator equals zero, tangent is undefined.

11. State the formula for $\cos(\alpha+\beta)$.

    $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. Standard angle addition formula for cosine function.

12. Identify the correct identity for $\sin(\alpha+\beta)$ among sign choices in the middle term.

    $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. The correct sign in sine addition formula is positive.

13. State the product-to-sum style result: what is $\sin\alpha\cos\beta+\cos\alpha\sin\beta$ equal to?

    $\sin\alpha\cos\beta+\cos\alpha\sin\beta=\sin(\alpha+\beta)$. This expression equals the sine addition formula result.

14. State the result: what is $\cos\alpha\cos\beta-\sin\alpha\sin\beta$ equal to?

    $\cos\alpha\cos\beta-\sin\alpha\sin\beta=\cos(\alpha+\beta)$. This expression equals the cosine addition formula result.

15. State the even-odd identity for sine: what is $\sin(-\theta)$?

    $\sin(-\theta)=-\sin\theta$. Sine is an odd function: $\sin(-x) = -\sin(x)$.

16. State the double-angle identity for tangent: what is $\tan(2\theta)$?

    $\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}$. Double-angle formula derived from tangent addition formula.

17. State the even-odd identity for tangent: what is $\tan(-\theta)$?

    $\tan(-\theta)=-\tan\theta$. Tangent is an odd function: $\tan(-x) = -\tan(x)$.

18. State the formula for $\tan(\alpha-\beta)$.

    $\tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$. Standard angle subtraction formula for tangent function.

19. State the cofunction identity for cosine: what is $\cos\left(\frac{\pi}{2}-\theta\right)$?

    $\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta$. Cosine and sine are cofunctions differing by $\frac{\pi}{2}$.

20. State the formula for $\sin(\alpha+\beta)$.

    $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Standard angle addition formula for sine function.

21. State the even-odd identity for cosine: what is $\cos(-\theta)$?

    $\cos(-\theta)=\cos\theta$. Cosine is an even function: $\cos(-x) = \cos(x)$.

22. State the formula for $\cos(\alpha-\beta)$.

    $\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$. Standard angle subtraction formula for cosine function.

23. Identify the expression equivalent to $\cos(\beta-\alpha)$ in terms of $\cos(\alpha-\beta)$.

    $\cos(\beta-\alpha)=\cos(\alpha-\beta)$. Cosine is an even function, so $\cos(\beta-\alpha) = \cos(\alpha-\beta)$.

24. Identify the Pythagorean identity relating $\sin\theta$ and $\cos\theta$.

    $\sin^2\theta+\cos^2\theta=1$. Fundamental Pythagorean identity for unit circle.

25. State the result: what is $\cos\alpha\cos\beta+\sin\alpha\sin\beta$ equal to?

    $\cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\alpha-\beta)$. This expression equals the cosine subtraction formula result.

26. Identify the expression equivalent to $\tan(\beta-\alpha)$ in terms of $\tan(\alpha-\beta)$.

    $\tan(\beta-\alpha)=-\tan(\alpha-\beta)$. Tangent is an odd function, so $\tan(\beta-\alpha) = -\tan(\alpha-\beta)$.

27. What identity defines tangent in terms of sine and cosine?

    $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Fundamental quotient identity relating tangent to sine and cosine.

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