Practice Pythagorean Trigonometric Identities in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
The fundamental identity \sin^2\theta + \cos^2\theta = 1 and its derived forms: 1 + \tan^2\theta = \sec^2\theta and 1 + \cot^2\theta = \csc^2\theta.
On the unit circle, the point (\cos\theta, \sin\theta) is always at distance 1 from the origin. By the Pythagorean theorem, x^2 + y^2 = 1 becomes \cos^2\theta + \sin^2\theta = 1. This single fact—that sine and cosine are tied to a circle—generates all three Pythagorean identities. Dividing through by \cos^2\theta or \sin^2\theta produces the other two forms.
Example 1
easyIf \sin(\theta) = \frac{3}{5} and \theta is in Quadrant I, find \cos(\theta) using the Pythagorean identity.
Example 2
mediumSimplify the expression \frac{1 - \cos^2(\theta)}{\sin(\theta) \cos(\theta)}.
Example 3
mediumProve that \tan^2(\theta) + 1 = \sec^2(\theta).
Example 4
hardSimplify \frac{\sec^2(\theta) - 1}{\csc^2(\theta) - 1}.