Trigonometric Identities & Equations

Pythagorean Identities

Using the Pythagorean identities to simplify trigonometric expressions.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

An Identity Hiding in the Unit Circle's Own Definition

Recall that (cos θ, sin θ) is defined as a point on the unit circle — a circle of radius 1 centered at the origin. Before reading on, think about what equation every point on that circle must satisfy, and what that means once (cos θ, sin θ) is substituted in for (x, y).

Every point on the unit circle satisfies x² + y² = 1, since the circle's equation directly states that every point is exactly distance 1 from the origin. Substituting x = cos θ and y = sin θ gives the Pythagorean identity immediately — it isn't a separate trigonometric fact to prove, it's the unit circle's own defining equation, restated.

Definition — The Pythagorean Identities

sin²θ + cos²θ = 1. Dividing every term by cos²θ gives tan²θ + 1 = sec²θ. Dividing every term by sin²θ gives 1 + cot²θ = csc²θ. (Recall sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ.)

Worked Example — Finding a Trig Value Using the Identity

If sin θ = 3/5 and θ is in Quadrant II, find cos θ. Using sin²θ + cos²θ = 1: cos²θ = 1 − 9/25 = 16/25, so cos θ = ±4/5. Since θ is in Quadrant II, where cosine is negative: cos θ = −4/5.

Worked Example — Verifying a Derived Identity Numerically

Using sin θ = 3/5 and cos θ = −4/5 from the previous example, verify tan²θ + 1 = sec²θ. tan θ = sin θ/cos θ = (3/5)/(−4/5) = −3/4, so tan²θ = 9/16, and tan²θ + 1 = 25/16. Separately, sec θ = 1/cos θ = 1/(−4/5) = −5/4, so sec²θ = 25/16. Both sides equal 25/16, confirming the identity.

Tip

When solving for a trig value using the Pythagorean identity, the square root always leaves a ± — resolve the sign using which quadrant the angle is in, never by default assuming the positive root.

Common Mistakes

Taking only the positive square root when solving for a trig value using the Pythagorean identity, without checking the quadrant.
√(cos²θ) = |cos θ|, giving both a positive and negative possibility — use the given quadrant to determine which sign is actually correct.
Treating sin²θ + cos²θ = 1 as if it applied to sin θ + cos θ directly, without the squares.
The identity specifically involves the squares of sine and cosine — sin θ + cos θ (unsquared) has no fixed value and isn't equal to 1 in general.

Key Takeaways

sin²θ + cos²θ = 1 comes directly from the unit circle's own defining equation, x² + y² = 1.
Dividing this identity by cos²θ or sin²θ derives the two related identities involving secant and cosecant, or cosecant and cotangent.
Solving for a trig value using this identity requires resolving a ± sign using the angle's quadrant.

Summary

The Pythagorean identities all trace back to a single geometric fact about the unit circle. The next lesson derives formulas for the sine and cosine of a sum or difference of two angles.

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