Question

Cot-1 ( √1+sinx +√ 1-sinx / √1+sinx -√ 1-sinx )= x/2 , x∈(0, π/4)

Answer 1

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Answer 2

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