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Chapter 5: Problem 60

Verify each identity. $$\frac{\sin x+\cos x}{\sin x}-\frac{\cos x-\sin x}{\cos x}=\sec x \csc x$$

Short Answer

Expert verified
After simplifying both sides, the left side is \(1 - 2\cot x\) and the right side is \(1 - 2\cot x\). Hence, the given equation is an identity.

Step by step solution

01

Rewrite Secant and Cosecant

Express the right side of the equation in terms of sine and cosine. The secant of \(x\) (\(\sec x\)) is equivalent to \(1 / \cos x\), and the cosecant of \(x\) (\(\csc x\)) is equivalent to \(1 / \sin x\). So, right side = \(\sec x \csc x = (1 / \cos x)*(1 / \sin x)\).
02

Simplify Right Side

Combine the denominators on the right side, which gives \(1 / (\cos x \sin x)\).
03

Simplify the Left Side

The left side of the equation is \((\sin x + \cos x) / \sin x - (\cos x - \sin x) / \cos x\). To simplify this, find the common denominator, which is \(\sin x \cos x\). Then, multiply and subtract to get \((\sin x^2 - \cos x^2) / (\sin x \cos x)\).
04

Apply the Pythagorean Identity

On simplifying the left side, use the Pythagorean Identity: \(\sin^2 x + \cos^2 x = 1\). Rewrite \(\sin^2 x\) as \(1 - \cos^2 x\), then substitute into the left side expression. This gives us \((1 - \cos^2 x - \cos^2 x) / (\sin x \cos x) = (1 - 2\cos^2 x) / (\sin x \cos x)\).
05

Rearrange Left Side

Rearrange the expression on left side, isolate \(\sin x \cos x\) in denominator, this gives us \(1 / (\sin x \cos x) - 2\cos^2 x / (\sin x \cos x)\). Here second term, \(2\cos^2x / \sin x \cos x\) simplifies to \(2\cos x / \sin x\), this simplification is valid because \(\sin x\) and \(\cos x\) are non-zero.
06

Apply Identity

Apply the identity \(\cot x = \cos x / \sin x\) on \(2\cos x / \sin x\) which gives \(2\cot x\). Also, \(\sin x\cos x\) in the denominator of the first term simplifies to \(1\) using identities, which leaves \(1 - 2\cot x\). As \(\cot x = 1 / \tan x\), the left side reduces to \(1 - 2 / \tan x\). Given \(1 / \tan x = \cot x\), the left side equals to \(1 - 2\cot x\).

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