Question
Differentiate tan -1 [sin x - cos x /sin x + cos x]
with respect to 1/2 x.
Answer :
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Let's solve this step by step, carefully. The problem is: Differentiate: $$ y = \tan^{-1} \left( \frac{\sin x - \cos x}{\sin x + \cos x} \right) $$ with respect to $\frac{1}{2} x$. --- ### Step 1: Simplify the differentiation variable We are asked for: $$ \frac{dy}{d(\frac{x}{2})} = \frac{dy}{dx} \cdot \frac{dx}{d(x/2)} $$ Since $\frac{dx}{d(x/2)} = 2$, we have: $$ \frac{dy}{d(x/2)} = 2 \frac{dy}{dx} $$ So first we differentiate $y$ w\.r.t $x$, then multiply by 2. --- ### Step 2: Simplify the argument
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