Pythagorean Trigonometric Identities Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Pythagorean Trigonometric Identities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: These identities express the geometric constraint that sine and cosine represent coordinates on a unit circle. They allow you to convert between trig functions and simplify expressions.

Common stuck point: When using \cos^2\theta = 1 - \sin^2\theta to find cosine from sine, remember to consider the \pm sign—the quadrant determines which sign is correct.

Sense of Study hint: Simplify by replacing sin^2 with 1 - cos^2 (or vice versa) to get everything in terms of one trig function. Then solve the resulting equation.

Worked Examples

Example 1

easy
If \sin(\theta) = \frac{3}{5} and \theta is in Quadrant I, find \cos(\theta) using the Pythagorean identity.

Solution

  1. 1
    Start with the Pythagorean identity: \sin^2(\theta) + \cos^2(\theta) = 1.
  2. 2
    Substitute \sin(\theta) = \frac{3}{5}: \left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1, so \frac{9}{25} + \cos^2(\theta) = 1.
  3. 3
    Solve: \cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25}, so \cos(\theta) = \pm\frac{4}{5}.
  4. 4
    Since \theta is in Quadrant I, \cos(\theta) > 0, so \cos(\theta) = \frac{4}{5}.

Answer

\cos(\theta) = \frac{4}{5}
The Pythagorean identity \sin^2\theta + \cos^2\theta = 1 directly relates sine and cosine. When you know one and the quadrant, you can find the other. The quadrant determines the sign of the result.

Example 2

medium
Simplify the expression \frac{1 - \cos^2(\theta)}{\sin(\theta) \cos(\theta)}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Prove that \tan^2(\theta) + 1 = \sec^2(\theta).

Example 2

hard
Simplify \frac{\sec^2(\theta) - 1}{\csc^2(\theta) - 1}.
← Back to Pythagorean Trigonometric Identities Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

trigonometric functionspythagorean theoremunit circle