Question

Prove that:
\[ \frac{\sec^3 0}{\sec^2 0 -1} + \frac{\cosec^3 0}{\cosec^2 0 -1} = \sec 0 \cosec 0 ( \sec 0 + \cosec 0 ). \]

Hint:

Use fundamental trigonometric identities to simplify complex expressions step by step.

Answer 1

Step 1: Express in terms of trigonometric identities We know:
\[ \sec^2 \theta - 1 = \tan^2 \theta, \quad \cosec^2 \theta - 1 = \cot^2 \theta. \] Rewriting each term:
\[ \frac{\sec^3 \theta}{\sec^2 \theta -1} = \frac{\sec^3 \theta}{\tan^2 \theta} = \sec \theta \cdot \frac{1}{\cos \theta}. \] Similarly,
\[ \frac{\cosec^3 \theta}{\cosec^2 \theta -1} = \frac{\cosec^3 \theta}{\cot^2 \theta} = \cosec \theta \cdot \frac{1}{\sin \theta}. \] Step 2: Adding both expressions
\[ \sec \theta \cdot \frac{1}{\cos \theta} + \cosec \theta \cdot \frac{1}{\sin \theta}. \] \[ = \sec \theta \cosec \theta (\sec \theta + \cosec \theta). \] Thus, the given equation is proved.