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Chapter 5: Problem 33

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

Short Answer

Expert verified
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 by step solution

01

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 - \( \frac{1}{\cos^{2} x} + \frac{1}{\sin^{2} x} \)
02

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.
03

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.

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