Problem 23 - Precalculus Essentials

Question

Verify each identity. $$\frac{1-\cos \theta}{\sin \theta}=\csc \theta-\cot \theta$$

Step-by-Step Solution

Answer

The identity \(\frac{1-\cos \theta}{\sin \theta} = \csc \theta - \cot \theta\) is true as both sides simplify to the same expression.

Step 1: Start with the Left Hand Side

The left-hand side (LHS) of the identity is \(\frac{1-\cos \theta}{\sin \theta}\). This cannot be simplified any further.

Step 2: Transform the Right Hand Side to match the LHS

The right-hand side (RHS) of the identity is \( \csc \theta - \cot \theta = \frac{1}{\sin\theta} - \frac{\cos\theta}{\sin\theta}\). This simplifies to\(\frac{1-\cos\theta}{\sin\theta}\).

Step 3: Verify the Identity

Since both the LHS and the RHS simplify to the same expression, the identity \(\frac{1-\cos \theta}{\sin \theta} = \csc \theta - \cot \theta\) is verified.