Problem 44 - Precalculus Essentials

Question

Find the exact value of each expression. Do not use a calculator. $$\frac{1}{\cot \frac{\pi}{4}}-\frac{2}{\csc \frac{\pi}{6}}$$

Step-by-Step Solution

Answer

The exact value of the expression is 2.

Step 1: Convert cotangent to tangent

Cotangent is the reciprocal of tangent. Based on this definition, the first term \(\frac{1}{\cot \frac{\pi}{4}}\) can be converted to \( \tan\frac{\pi}{4} \). The tangent of 45 degrees \(\frac{\pi}{4}\) radians is 1, so this term simplifies to 1.

Step 2: Convert cosecant to sine

Cosecant is the reciprocal of sine. Hence, the second term \( -\frac{2}{\csc \frac{\pi}{6}} \) can be changed to \( -2 \cdot \sin\frac{\pi}{6} \). The sine of 30 degrees \(\frac{\pi}{6}\) radians is 0.5, so this term simplifies to -1.

Step 3: Subtract the values

Now, the expression becomes \(1 - (-1)\), which simplifies to 2.