Question

Prove the following
$\sf \cot ^{ - 1} \left \{ \dfrac{ \sqrt{1 + \sin \: x } \: \: + \sqrt{1 - \sin \: x } }{ \sqrt{1 + \sin \: x} \: \: - \sqrt{1 - \sin \: x } } \right \} \: = \dfrac{x}{2} ; \: x \in \: \bigg(0, \dfrac{ \pi}{4} \bigg)$
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Answer 1

Answer:

Correct option is

D

π−2x

We have,

⇒cot−1(1−sinx−1+sinx1−sinx+1+sinx)

⇒cot−1((1−sinx)−(1+sinx)(1−sinx+1+sinx)2)

⇒cot−1(1−sinx−1−sinx1−sinx+1+sinx+21−sin2x)

⇒cot−1(−2sinx2+2cos2x)

⇒cot−1(−sinx1+cosx)

⇒cot−1

Answer 2

Answer:

We have,

⇒cot−1(1−sinx−1+sinx1−sinx+1+sinx)

⇒cot−1((1−sinx)−(1+sinx)(1−sinx+1+sinx)2)

⇒cot−1(1−sinx−1−sinx1−sinx+1+sinx+21−sin2x)

⇒cot−1(−2sinx2+2cos2x)

⇒cot−1(−sinx1+cosx)

⇒cot−1⎝⎛−2sin2