Question

cot⁻¹(√1+sinx+√1-sinx/√1+sinx-√1-sinx) = x/2,0 < x < π/2,prove it

Answer 1

$\textbf{Check The Attachment}$

$\textbf{The Identities Used}$>>>

✔$\upper{sin}^{2}$(x) + $\upper{cos}^{2}$(x) = 1

✔sin(2x) = 2sin(x)cos(x)

✔$\frac{cos(x)}{sin(x)}$ = cot(x)

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Answer 2

${cot}^{ - 1} ( \frac{ \sqrt{1 + sinx} + \sqrt{1 - sinx} }{ \sqrt{1 + sinx} - \sqrt{1 - sinx} } ) \\ \\ = {cot}^{ - 1} ( \frac{cos \frac{x}{2} + sin\frac{x}{2} + cos \frac{x}{2} - sin \frac{x}{2} }{cos \frac{x}{2} + sin \frac{x}{2} - cos \frac{x}{2} + sin \frac{x}{2} } ) \\ \\ = {cot}^{ - 1} ( \frac{2cos \frac{x}{2} }{2sin \frac{x}{2} } ) \\ \\ = {cot}^{ - 1} cot \frac{x}{2} \\ \\ = \frac{x}{2}$