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Chapter 4: Problem 38

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

Short Answer

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

Step by step solution

01

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 \(\sin^2(\theta)\).
02

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 \(1 + \cot^2(\frac{\pi}{6}) - \cot^2(\frac{\pi}{6})\)
03

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.

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