Problem 38 - Precalculus Essentials

Question

Verify each identity. $$\frac{\tan ^{2} x-\cot ^{2} x}{\tan x+\cot x}=\tan x-\cot x$$

Step-by-Step Solution

Answer

The identity \(\frac{\tan^{2} x - \cot^{2} x}{\tan x + \cot x} = \tan x - \cot x\) is verified.

Step 1: Write the Definitions

Rewrite the given expression with the definitions of \( \tan x \) and \( \cot x \). Remember, \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). The expression becomes \(\frac{(\frac{\sin^{2} x}{\cos^{2} x} - \frac{\cos^{2} x}{\sin^{2} x})}{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}} = \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}\).

Step 2: Simplify the Expression

Simplify the numerator and the denominator of the fraction in the left-hand side (LHS). The simplification gives: \(\frac{\sin^{4} x - \cos^{4} x}{\sin^{2} x \cos^{2} x}\div \frac{\sin^{2} x + \cos^{2} x}{\sin x \cos x}\).

Step 3: Further Simplify the Expression

Further simplification gives: \(\frac{\sin^{4} x - \cos^{4} x}{\sin^{2} x + \cos^{2} x} = \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}\), which will be recognized as the difference of squares \(\sin^{4} x - \cos^{4} x =(\sin^{2} x + \cos^{2} x)(\sin^{2} x - \cos^{2} x)\). Then cancel out the common factors.

Step 4: Final Simplification

After cancellation, it is observed that, \(\frac{\sin^{2} x - \cos^{2} x}{\sin x \cos x} = \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}\). This implies \(\frac{\sin x}{\cos x} - \frac{\cos x}{\sin x} = \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}\), and thus the identity is verified.