Pythagorean Trigonometric Identities Math Example 2

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Example 2

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

Solution

  1. 1
    Recognize that 1−cos⁡2(θ)=sin⁡2(θ)1 - \cos^2(\theta) = \sin^2(\theta) by the Pythagorean identity.
  2. 2
    Substitute: sin⁡2(θ)sin⁡(θ)cos⁡(θ)\frac{\sin^2(\theta)}{\sin(\theta) \cos(\theta)}.
  3. 3
    Cancel sin⁡(θ)\sin(\theta): sin⁡(θ)cos⁡(θ)=tan⁡(θ)\frac{\sin(\theta)}{\cos(\theta)} = \tan(\theta).

Answer

tan⁡(θ)\tan(\theta)
The Pythagorean identity allows you to replace 1−cos⁡2θ1 - \cos^2\theta with sin⁡2θ\sin^2\theta (or 1−sin⁡2θ1 - \sin^2\theta with cos⁡2θ\cos^2\theta). This substitution is one of the most common simplification techniques in trigonometry.

About Pythagorean Trigonometric Identities

The fundamental identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 and its derived forms: 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta and 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta.

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More Pythagorean Trigonometric Identities Examples

Example 1 easy

If [formula] and [formula] is in Quadrant I, find [formula] using the Pythagorean identity.

Example 3 medium

Prove that [formula].

Example 4 hard

Simplify [formula].

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