Problem 37 - Precalculus Essentials

Question

Use an identity to find the value of each expression. Do not use a calculator. $$\sec ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{3}$$

Step-by-Step Solution

Answer

The value of the expression \( \sec ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{3} \) is 1.

Step 1: Recognize the identity

The given expression \( \sec ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{3} \) resembles the Pythagorean identity \( \sec^{2} \phi - \tan^{2} \phi = 1 \). So using this identity, we can say that the given expression equals 1 regardless of the angle.

Step 2: Apply the identity

The given expression is \( \sec ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{3} \), using the Pythagorean identity discussed before, where \( \phi = \frac{\pi}{3} \), the result is 1.