Problem 60 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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 = \(\sec x \csc x = (1 / \cos x)*(1 / \sin x)\).

Step 2: Simplify Right Side

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

Step 3: 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 \((\sin x^2 - \cos x^2) / (\sin x \cos x)\).

Step 4: 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 \((1 - \cos^2 x - \cos^2 x) / (\sin x \cos x) = (1 - 2\cos^2 x) / (\sin x \cos x)\).

Step 5: 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.

Step 6: 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 \(1 - 2\cot x\).