Question

Let $y = \tan^{-1}\left(\frac{\sin x + \cos x}{\sin x - \cos x}\right), \; 0<x < \frac{\pi}{2}$

• A. X
• B. -1
• C. -X
• D. 2x
• E. -2x

Hint:

Math Tip: The expression $\frac{\cos x + \sin x}{\cos x - \sin x}$ always simplifies to $\tan(\frac{\pi}{4} + x)$. Similarly, $\frac{\cos x - \sin x}{\cos x + \sin x}$ simplifies to $\tan(\frac{\pi}{4} - x)$. Memorizing these two transformations is a major time-saver for integration and differentiation.

Answer 1

The Correct Option is B

Concept:
Calculus - Differentiation of Inverse Trigonometric Functions.
Always simplify the trigonometric expression inside the inverse function before attempting to differentiate.
Step 1: Simplify the inner expression.
Given the inner fraction: $\frac{\sin x + \cos x}{\sin x - \cos x}$. Divide both the numerator and the denominator by $\cos x$: $$ \frac{\frac{\sin x}{\cos x} + \frac{\cos x}{\cos x}}{\frac{\sin x}{\cos x} - \frac{\cos x}{\cos x}} = \frac{\tan x + 1}{\tan x - 1} $$
Step 2: Format the expression to match a trigonometric identity.
Factor out a negative sign from the denominator to match the standard tangent addition formula: $$ = \frac{1 + \tan x}{-(1 - \tan x)} = -\left(\frac{1 + \tan x}{1 - \tan x}\right) $$
Step 3: Apply the tangent addition formula.
Recall that $\tan(\frac{\pi}{4}) = 1$. Substitute this into the expression: $$ -\left(\frac{\tan(\frac{\pi}{4}) + \tan x}{1 - \tan(\frac{\pi}{4})\tan x}\right) $$ This is the exact expansion of $\tan(A + B)$: $$ = -\tan\left(\frac{\pi}{4} + x\right) $$
Step 4: Absorb the negative sign.
Since the tangent function is odd ($\tan(-\theta) = -\tan\theta$), move the negative sign inside the argument: $$ = \tan\left(-\frac{\pi}{4} - x\right) $$
Step 5: Evaluate the inverse function.
Substitute the simplified expression back into the original function: $$ y = \tan^{-1}\left[ \tan\left(-\frac{\pi}{4} - x\right) \right] $$ Because the inverse and original functions cancel each other out: $$ y = -\frac{\pi}{4} - x $$
Step 6: Differentiate the simplified function.
Now, differentiate $y$ with respect to $x$: $$ \frac{dy}{dx} = \frac{d}{dx}\left(-\frac{\pi}{4} - x\right) $$ Since $-\frac{\pi}{4}$ is a constant, its derivative is $0$: $$ \frac{dy}{dx} = -1 $$