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Chapter 5: Problem 17

Verify the identity. $$\frac{\tan ^{2} \theta}{\sec \theta}=\sin \theta \tan \theta$$

Short Answer

Expert verified
Verified, the identity \(\frac{\tan ^{2} \theta}{\sec \theta}=\sin \theta \tan \theta\) is true.

Step by step solution

01

Convert into fundamental trigonometric ratios

Convert \(\tan ^{2} \theta\) and \(\sec \theta\) into sine and cosine. \(\tan ^{2} \theta = \sin ^{2} \theta / \cos ^{2} \theta\) and \(\sec \theta = 1 / \cos \theta\)
02

Substitute the conversions

Substitute the conversions into the left-hand side of the identity, \(\frac{\sin ^{2} \theta / \cos ^{2} \theta}{1 / \cos \theta}\)
03

Simplify expression

Simplify the expression which results to \(\sin \theta \cdot \sin \theta / \cos \theta\) = \(\sin \theta \tan \theta\)
04

Conclude the proof

Upon simplification, the left-hand side of the identity becomes \(\sin \theta \tan \theta\), which is the same as the right-hand side. Therefore, the identity is verified.

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