Problem 11 - Precalculus Essentials

Question

Verify each identity. $$\csc \theta-\sin \theta=\cot \theta \cos \theta$$

Step-by-Step Solution

Answer

The given trigonometric identity \(\csc \theta - \sin \theta = \cot \theta \cos \theta\) is verified.

Step 1: Break down trigonometric identities

Express \(\csc \theta\) and \(\cot \theta\) in terms of sine and cosine. \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

Step 2: Substitute the identities in the equation

Replace \(\csc \theta\) and \(\cot \theta\) in the equation with their sine and cosine equivalents: \(\frac{1}{\sin \theta}-\sin \theta=\frac{\cos \theta}{\sin \theta}\cos \theta\)

Step 3: Simplify the equation

Clear the denominators by multiplying through by \(\sin \theta\) to yield: \(1 - (\sin \theta)^2 = \cos ^{2} \theta \).

Step 4: Use Pythagorean identity

Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Rearrange it to get \(\sin^2 \theta = 1 - \cos^2 \theta\). Substitute this into the equation from Step 3 to yield: \(1 - (1 - \cos^2 \theta) = \cos^2 \theta \). Simplify to get \( \cos^2 \theta = \cos^2 \theta \).