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Proving the
Pythagorean Identity

Master the proof and application of \(\sin^2\theta + \cos^2\theta = 1\), and learn to find trigonometric values using this fundamental identity.

Trigonometry

The Unit Circle Foundation

Every point on the unit circle (a circle with radius 1 centered at the origin) can be written as \((\cos\theta, \sin\theta)\). This simple fact is the key to one of the most important identities in all of mathematics.

\(\theta\) \(\cos\theta\) \(\sin\theta\) \(r = 1\) \((\cos\theta,\,\sin\theta)\) \(x\) \(y\) \(1\) \(-1\) \(1\) \(-1\)

From Pythagoras to Trig

Recall the Pythagorean Theorem: in a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), we have \(a^2 + b^2 = c^2\). On the unit circle, the right triangle formed by any point has:

Horizontal leg
\(\cos\theta\) (the x-coordinate)
Vertical leg
\(\sin\theta\) (the y-coordinate)
Hypotenuse
\(r = 1\) (the radius)

Why This Identity Matters

The Pythagorean Identity is not just a formula to memorize: it connects geometry, algebra, and trigonometry into a single powerful tool. Here is what it unlocks:

1
Find missing trig values: Given \(\sin\theta\), you can find \(\cos\theta\) (and vice versa).
2
Simplify expressions: Replace \(\sin^2\theta + \cos^2\theta\) with 1, or rearrange to substitute parts.
3
Prove other identities: The two derived identities \(\tan^2\theta + 1 = \sec^2\theta\) and \(1 + \cot^2\theta = \csc^2\theta\) both come from dividing this one.
4
Determine quadrant signs: Combined with quadrant information, you can resolve the \(\pm\) sign.

The Formal Proof

1
Start with the unit circle. Let \(P = (\cos\theta, \sin\theta)\) be any point on the circle of radius 1 centered at the origin.
2
Write the equation of the circle. The unit circle is defined by \(x^2 + y^2 = 1\).
3
Substitute the coordinates. Since \(x = \cos\theta\) and \(y = \sin\theta\), we get \(\cos^2\theta + \sin^2\theta = 1\).
4
This holds for all \(\theta\). Every angle places \(P\) on the circle, so the identity is always true. \(\blacksquare\)

Visualizing the Proof

\(\theta\) \(\cos\theta\) \(\sin\theta\) \(1\) Pythagorean Thm \(a^2 + b^2 = c^2\) \(\cos^2\!\theta + \sin^2\!\theta = 1\) \(x\) \(y\)

The Three Pythagorean Identities

By dividing \(\sin^2\theta + \cos^2\theta = 1\) by either \(\cos^2\theta\) or \(\sin^2\theta\), we derive two more identities:

Primary
\(\sin^2\theta + \cos^2\theta = 1\)
Divide by \(\cos^2\theta\)
\(\tan^2\theta + 1 = \sec^2\theta\)
Divide by \(\sin^2\theta\)
\(1 + \cot^2\theta = \csc^2\theta\)

Identity vs. Equation

Students often confuse identities with equations. Understanding the difference is critical for proofs and problem solving.

Identity Equation
Definition True for all values of the variable True for specific values only
Example \(\sin^2\theta + \cos^2\theta = 1\) \(\sin\theta = \tfrac{1}{2}\)
You can... Substitute it into any expression Solve it for specific \(\theta\)
Goal Prove it is always true Find which \(\theta\) make it true

Quadrant Sign Rules

The identity tells us the magnitude of \(\sin\theta\) or \(\cos\theta\), but the quadrant tells us the sign. Memorize which functions are positive in each quadrant:

Q I All positive \(\sin\theta+, \cos\theta+, \tan\theta+\) A Q II Sine positive \(\sin\theta+, \cos\theta-, \tan\theta-\) S Q III Tangent positive \(\sin\theta-, \cos\theta-, \tan\theta+\) T Q IV Cosine positive \(\sin\theta-, \cos\theta+, \tan\theta-\) C

How to Apply the Identity

Follow these steps whenever you need to find a missing trig value from a known one:

1
Write the identity: \(\sin^2\theta + \cos^2\theta = 1\).
2
Substitute the known value (e.g., \(\sin\theta = \tfrac{3}{5}\)).
3
Solve for the unknown squared: \(\cos^2\theta = 1 - \sin^2\theta\).
4
Take the square root: \(\cos\theta = \pm\sqrt{1 - \sin^2\theta}\).
5
Use the quadrant to choose the correct sign (\(+\) or \(-\)).

Example 1: Finding \(\cos\theta\)

1
Write the identity: \(\sin^2\theta + \cos^2\theta = 1\)
2
Substitute: \(\left(\dfrac{3}{5}\right)^2 + \cos^2\theta = 1\)
3
Simplify: \(\dfrac{9}{25} + \cos^2\theta = 1 \implies \cos^2\theta = \dfrac{16}{25}\)
4
Solve: \(\cos\theta = \pm\dfrac{4}{5}\). Since Q I, cosine is positive: \(\cos\theta = \dfrac{4}{5}\)

Example 2: Finding \(\sin\theta\) in Q III

1
Substitute: \(\sin^2\theta + \left(-\dfrac{5}{13}\right)^2 = 1\)
2
Simplify: \(\sin^2\theta + \dfrac{25}{169} = 1 \implies \sin^2\theta = \dfrac{144}{169}\)
3
Solve: \(\sin\theta = \pm\dfrac{12}{13}\). In Q III, sine is negative: \(\sin\theta = -\dfrac{12}{13}\)

Example 3: Given \(\tan\theta\)

1
Use \(\tan^2\theta + 1 = \sec^2\theta\): \(\left(-\dfrac{7}{24}\right)^2 + 1 = \sec^2\theta\)
2
Compute: \(\dfrac{49}{576} + 1 = \dfrac{625}{576} \implies \sec\theta = \pm\dfrac{25}{24}\)
3
Q II: cosine is negative, so \(\sec\theta = -\dfrac{25}{24}\), giving \(\cos\theta = -\dfrac{24}{25}\)
4
Find sine: \(\sin\theta = \tan\theta \cdot \cos\theta = \left(-\dfrac{7}{24}\right)\left(-\dfrac{24}{25}\right) = \) \(\dfrac{7}{25}\)

Rearranged Forms

Depending on what you need to isolate, rearrange the identity into one of these equivalent forms:

Solve for \(\sin^2\theta\)
\(\sin^2\theta = 1 - \cos^2\theta\)
Solve for \(\cos^2\theta\)
\(\cos^2\theta = 1 - \sin^2\theta\)

Common Mistakes

✕ Forgetting the \(\pm\)
When you take a square root, always write \(\pm\). Then use the quadrant to choose the correct sign.
✕ Squaring incorrectly
Remember: \(\sin^2\theta\) means \((\sin\theta)^2\), not \(\sin(\theta^2)\). Square the entire function value.
✕ Wrong quadrant sign
Always check: which trig functions are positive in your quadrant? A wrong sign flips the entire answer.

Summary of Pythagorean Identities

Identity Formula Derived By
Primary \(\sin^2\theta + \cos^2\theta = 1\) Unit circle definition
Tangent form \(\tan^2\theta + 1 = \sec^2\theta\) Divide by \(\cos^2\theta\)
Cotangent form \(1 + \cot^2\theta = \csc^2\theta\) Divide by \(\sin^2\theta\)

Question 1 of 5

If \(\sin\theta = \dfrac{5}{13}\) and \(\theta\) is in Quadrant II, what is \(\cos\theta\)?

Question 2 of 5

Which of the following is a valid rearrangement of the Pythagorean Identity?

Question 3 of 5

If \(\cos\theta = \dfrac{7}{25}\) and \(\theta\) is in Quadrant IV, what is \(\tan\theta\)?

Question 4 of 5

The identity \(\tan^2\theta + 1 = \sec^2\theta\) is derived from the primary Pythagorean Identity by:

Question 5 of 5

Simplify the expression: \(\dfrac{\sin^2\theta}{1 - \cos\theta}\), given that \(\cos\theta \neq 1\).