Pythagorean Trigonometric Identities Math Example 4

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Example 4

hard
Simplify sec⁡2(θ)−1csc⁡2(θ)−1\frac{\sec^2(\theta) - 1}{\csc^2(\theta) - 1}.

Solution

  1. 1
    Use the Pythagorean identities: sec⁡2θ−1=tan⁡2θ\sec^2\theta - 1 = \tan^2\theta and csc⁡2θ−1=cot⁡2θ\csc^2\theta - 1 = \cot^2\theta.
  2. 2
    So the expression becomes tan⁡2θcot⁡2θ=tan⁡2θ⋅tan⁡2θ=tan⁡4θ\frac{\tan^2\theta}{\cot^2\theta} = \tan^2\theta \cdot \tan^2\theta = \tan^4\theta.

Answer

tan⁡4(θ)\tan^4(\theta)
Using the Pythagorean identities sec⁡2θ=1+tan⁡2θ\sec^2\theta = 1 + \tan^2\theta and csc⁡2θ=1+cot⁡2θ\csc^2\theta = 1 + \cot^2\theta, we can replace the numerator and denominator. Since cot⁡θ=1tan⁡θ\cot\theta = \frac{1}{\tan\theta}, dividing by cot⁡2θ\cot^2\theta is the same as multiplying by tan⁡2θ\tan^2\theta.

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.

Learn more about Pythagorean Trigonometric Identities →

More Pythagorean Trigonometric Identities Examples

Example 1 easy

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

Example 2 medium

Simplify the expression [formula].

Example 3 medium

Prove that [formula].

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