Problem 21 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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 \( \frac{\sec^{2} t - 1}{\frac{1}{\cos t}} \).

Step 2: 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 \( \sec t - \cos t \).

Step 3: Verify the identity

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