Question

find the principal value of cot^-1[ root 1-sin π/2+ root 1+sin π/2 / root1- sin π/2 + root1 +π/2

Answer 1

SOLUTION

TO DETERMINE

The value of

$\displaystyle \sf{ { \cot}^{ - 1} \bigg( \: \frac{ \sqrt{1 - \sin x} + \sqrt{1 + \sin x} }{ \sqrt{1 - \sin x} - \sqrt{1 + \sin x}} \: \bigg) }$

EVALUATION

$\displaystyle \sf{ { \cot}^{ - 1} \bigg( \: \frac{ \sqrt{1 - \sin x} + \sqrt{1 + \sin x} }{ \sqrt{1 - \sin x} - \sqrt{1 + \sin x}} \: \bigg) }$

$\displaystyle \sf{ = { \cot}^{ - 1} \bigg( \: \frac{ \sqrt{ { \cos}^{2} \frac{x}{2} + { \sin}^{2} \frac{x}{2} - 2 \sin \frac{x}{2} \cos \frac{x}{2} } + \sqrt{ { \cos}^{2} \frac{x}{2} + { \sin}^{2} \frac{x}{2} + 2 \sin \frac{x}{2} \cos \frac{x}{2}} }{ \sqrt{ { \cos}^{2} \frac{x}{2} + { \sin}^{2} \frac{x}{2} - 2 \sin \frac{x}{2} \cos \frac{x}{2} } - \sqrt{ { \cos}^{2} \frac{x}{2} + { \sin}^{2} \frac{x}{2} + 2 \sin \frac{x}{2} \cos \frac{x}{2}}} \: \bigg) }$

$\displaystyle \sf{ = { \cot}^{ - 1} \bigg[\: \frac{ \sqrt{{ \bigg( \cos \frac{x}{2} - \sin \frac{x}{2} \bigg)}^{2} } + \sqrt{ { \bigg( \cos \frac{x}{2} + \sin \frac{x}{2} \bigg)}^{2}} }{ \sqrt{{ \bigg( \cos \frac{x}{2} - \sin \frac{x}{2} \bigg)}^{2}} - \sqrt{ { \bigg( \cos \frac{x}{2} - \sin \frac{x}{2} \bigg)}^{2}}} \bigg] }$

$\displaystyle \sf{ = { \cot}^{ - 1} \bigg[\: \frac{ 2 \cos \frac{x}{2} }{ - 2 \sin \frac{x}{2} }^{} \bigg] }$

$\displaystyle \sf{ = { \cot}^{ - 1} \bigg[\: - \cot \frac{x}{2} \bigg] }$

$\displaystyle \sf{ = { \cot}^{ - 1} \bigg[\: \cot \bigg(\pi - \frac{x}{2} \bigg) \bigg] }$

$\displaystyle \sf{ = \pi - \frac{x}{2} }$

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