MCQ
$\frac{d}{{dx}}\left[ {\log \left( {1 + \sin x} \right) + \log \sec {{\left( {\frac{\pi }{4} - \frac{x}{2}} \right)}^2}} \right] = .....$
- ✓$0$
- B$4\left( {\frac{{\cos x - \tan x}}{{\sin x + \cos x}}} \right)$
- C${\log _e}2$
- D$ - {\log _e}2$
$\frac{d}{dx}\left[=log(1+sinx)+log(sec(\frac{\pi}{4}-\frac{x}{2})^2\right]$
$=\left[\frac{1.cosx}{1+sinx}+2\frac{d}{dx}log(sec(\frac{\pi}{4}-\frac{x}{2}))\right]$
$=\frac{1.cosx}{1+sinx}-\frac{2}{2}\frac{-(1)sec(\frac{\pi}{4}-\frac{x}{2})tan(\frac{\pi}{4}-\frac{x}{2})}{sec(\frac{\pi}{4}-\frac{x}{2})}$
$=\frac{1.cosx}{1+sinx}-tan(\frac{\pi}{4}-\frac{x}{2})$
$=\frac{cos^2\frac{x}{2}-sin^2\frac{x}{2}}{(sin\frac{x}{2}+cos\frac{x}{2})^2}-\frac{cos\frac{x}{2}-sin\frac{x}{2}}{(sin\frac{x}{2}+cos\frac{x}{2})}$
$=\frac{cos^2\frac{x}{2}-sin^2\frac{x}{2}-cos^2\frac{x}{2}+sin^2\frac{x}{2}}{(sin\frac{x}{2}+cos\frac{x}{2})^2}$
$=0$
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