Question

ICBSE 2011,
SOLUTION We have
[
begin{aligned}
L H S &=cot ^{-1}left{frac{(sqrt{1+sin x}+sqrt{1-sin x})}{(sqrt{1+sin x}-sqrt{1-sin x})} times frac{(sqrt{1+sin x}+sqrt{1-sin x})}{(sqrt{1+sin x}+sqrt{1-sin x})}right.
&=cot ^{-1}left{frac{(1+sin x)+(1-sin x)+2 sqrt{1-sin ^{2} x}}{(1+sin x)-(1-sin x)}right}
=& cot ^{-1}left{frac{2(1+cos x)}{2 sin x}right}=cot ^{-1}left(frac{1+cos x}{sin x}right)
=operatorname{ten} cos x &left[frac{2 cos ^{2}left(frac{x}{2}right)}{2 sin left(frac{x}{2}right) cos left(frac{x}{2}right)}right}=cot ^{-1}left(cot frac{x}{2}right)=frac{x}{2}=mathrm{RHS}
end{aligned}
]

# EXAMPLE 32 Prove that cot-1 me that on-1 V1+ sin x + 11-sinx (V1+ sin x-v1 - sin x YE [CBSE 2011, '14)

Solution