Problem 38 - Precalculus Essentials

Question

Use an identity to find the value of each expression. Do not use a calculator. $$\csc ^{2} \frac{\pi}{6}-\cot ^{2} \frac{\pi}{6}$$

Step-by-Step Solution

Answer

The value of the expression \(\csc^2(\frac{\pi}{6}) - \cot^2(\frac{\pi}{6})\) is 1.

Step 1: Recall the Pythagorean Identity

This problem involves rearranging the Pythagorean identity for trigonometric functions, which is \( \csc^2(\theta) = 1 + \cot^2(\theta)\). This identity comes directly from the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\), dividing all terms by \(\sin^2(\theta)\).

Step 2: Use the Pythagorean Identity

Substitute \(\csc^2(\frac{\pi}{6})\) with the Pythagorean identity. The original equations transforms into \(1 + \cot^2(\frac{\pi}{6})\) . In the original equation, subtract \(\cot^2(\frac{\pi}{6})\) from both sides to obtain \(1 + \cot^2(\frac{\pi}{6}) - \cot^2(\frac{\pi}{6})\)

Step 3: Simplify the Expression

Finally, simplify \(1 + \cot^2(\frac{\pi}{6}) - \cot^2(\frac{\pi}{6})\) to just \(1\).This is because once you subtract \(\cot^2(\frac{\pi}{6})\) from both sides, it will cancel out, leaving you with 1 as your solution.