Question

cot[√1+sinx + √1-sinx / √1+sinx - √1-sinx]​

Answer 1

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