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Chapter 5: Problem 22

Verify the identity. $$\sec ^{6} x(\sec x \tan x)-\sec ^{4} x(\sec x \tan x)=\sec ^{5} x \tan ^{3} x$$

Short Answer

Expert verified
Through algebraic manipulation and applying the Pythagorean identity, we successfully simplify the left-hand side of the equation to match the right-hand side, thus verifying the identity.

Step by step solution

01

Simplify the left side

The left-hand side can be simplified by factoring out \(\sec^5x\tan x\). This gives \(\sec^5x\tan x(\sec x-\sec^2x)\).
02

Apply Trigonometric Identity

Replace \(\sec^2x\) in the above equation by \(1+\tan^2x\), based on the Pythagorean identity. The expression on the left side now becomes \(\sec^5x\tan x(\sec x - (1 + \tan^2x))\).
03

Simplify the Expression

Simplify the above expression to get \(\sec^5x\tan x(\sec x -1 - \tan^2x)\). By reordering terms, it simplifies to \(\sec^5x\tan^3x\), which is the expression on the right-hand side.

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