Pythagorean Trigonometric Identities Math Example 3

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Example 3

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

Solution

  1. 1
    Start with sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1 and divide every term by cos⁡2(θ)\cos^2(\theta).
  2. 2
    sin⁡2(θ)cos⁡2(θ)+cos⁡2(θ)cos⁡2(θ)=1cos⁡2(θ)\frac{\sin^2(\theta)}{\cos^2(\theta)} + \frac{\cos^2(\theta)}{\cos^2(\theta)} = \frac{1}{\cos^2(\theta)}, which gives tan⁡2(θ)+1=sec⁡2(θ)\tan^2(\theta) + 1 = \sec^2(\theta).

Answer

tan⁡2(θ)+1=sec⁡2(θ)(proven)\tan^2(\theta) + 1 = \sec^2(\theta) \quad \text{(proven)}
Dividing the fundamental Pythagorean identity by cos⁡2θ\cos^2\theta yields the tangent-secant form. Similarly, dividing by sin⁡2θ\sin^2\theta gives 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta. These are the three Pythagorean identities.

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 4 hard

Simplify [formula].

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