Problem 33 - Precalculus Essentials

Question

Verify each identity. $$\sec ^{2} x \csc ^{2} x=\sec ^{2} x+\csc ^{2} x$$

Step-by-Step Solution

Answer

The equation \( \sec^{2} x \csc^{2} x=\sec^{2} x+\csc^{2} x \) is indeed an identity and is proven by converting the sec and csc functions into their reciprocal identities and simplifying the equation.

Step 1: Apply the Reciprocal Identities

Rewrite \( \sec^{2} x \) and \( \csc^{2} x \) as \( \frac{1}{\cos^{2} x} \) and \( \frac{1}{\sin^{2} x} \) on the left side of the equation. So, It becomes \( \frac{1}{\cos^{2} x} \cdot \frac{1}{\sin^{2} x} \). On the right side rewrite \( \sec^{2} x \) and \( \csc^{2} x \) as \( \frac{1}{\cos^{2} x} \) and \( \frac{1}{\sin^{2} x} \) respectively. Hence the equation can be written as - \( \frac{1}{\cos^{2} x} + \frac{1}{\sin^{2} x} \)

Step 2: Simplify

As cotangent can be represented as \( \frac{\cos^{2} x}{\sin^{2} x} \), let's use cotangent to simplify the equation. We have: \( \frac{1}{\cos^{2} x} \cdot \frac{1}{\sin^{2} x} \) that equals \( \frac{1}{\cot^{2} x} = \cot^{-2} x \). Now we ready to check our identity.

Step 3: Check The Identity

The comparison of both sides of equation is now easier. Our left side \( \cot^{-2} x \) equals to the right side \( \frac{1}{\cos^{2} x} + \frac{1}{\sin^{2} x} \). The equation has therefore proved to be an identity.