Explanation

(a)
Let $y = \lim_{x \rightarrow \frac{\pi}{2}}\left( \sin x \right)^{\tan x}$
$\Rightarrow \ \ \ \ \ \log{y = \lim_{x \rightarrow \frac{\pi}{2}}{\tan{x\log{\sin x}}}}$
$= \lim_{x \rightarrow \frac{\pi}{2}}\frac{\log{\sin x}}{\cot x}$
$= \lim_{x \rightarrow \frac{\pi}{2}}\frac{\frac{1}{\sin x}.\cos x}{- cosec^{2}x}$ [by L’ Hospital’s rule]
$= 0$
$\Rightarrow \ \ \ \ \ y = e^{0} = 1$