/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Verify each identity. $$\frac{... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 21

Verify each identity. $$\frac{\tan ^{2} t}{\sec t}=\sec t-\cos t$$

Short Answer

Expert verified
By substituting the trigonometric identities and then simplifying the equation, it can be shown that \( \frac{\tan ^{2} t}{\sec t} \) is indeed equal to \( \sec t - \cos t \) proving the identity.

Step by step solution

01

Replace the tan and sec

We start by replacing \( \tan^{2} t \) with \( \sec^{2} t - 1 \) and \( \sec t \) with \( \frac{1}{\cos t} \). This gives us \( \frac{\sec^{2} t - 1}{\frac{1}{\cos t}} \).
02

Simplify the expression

Next, we simplify the expression by multiplying both the numerator and the denominator with \( \cos t \). This gives us \( (\sec^{2} t - 1)\cos t = \cos t \sec^{2} t - \cos t \). Remember that \( \sec t \) is equal to \( \frac{1}{\cos t} \) and therefore \( \cos t \sec^{2} t = \frac{\cos t}{\cos^{2} t} = \frac{1}{\cos t} = \sec t \). The expression becomes \( \sec t - \cos t \).
03

Verify the identity

We see that this is equal to the right hand side of the original equation, therefore proving the identity.

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