Question

please give me right answers with full method
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Answer 1

To simplify :

$\star \: { \tan}^{ - 1} \frac{1}{ \sqrt{ {x}^{2} - 1 } } \: \: \:$

Here ,

$let \: x = \sec \theta \: \\ or \: \ \\ \theta = { \sec}^{ - 1} x$

$\implies \: { \tan}^{ - 1} ( \frac{1}{ \sqrt{ { \sec}^{2} \theta - 1 } } ) \\ \\ \implies \: { \tan}^{ - 1} ( \frac{1}{ \sqrt{ { \tan}^{2} \theta } } ) \\ \\ \implies \: { \tan}^{ - 1} ( \frac{1}{ \tan \theta} ) \\ \\ \implies \: { \tan}^{ - 1} ( \cot \theta) \\ \\ \implies \: { \tan}^{ - 1} ( \tan( \frac{\pi}{2} - \theta)) \\ \\ \implies \: \frac{\pi}{2} - \theta \\ \\ \implies \: \frac{\pi}{2} - { \sec}^{ - 1} x$

Formula used :

$\star \boxed{\bold{ { \sec }^{2} \theta - { \tan}^{2} \theta = 1}} \\ \\ \star \boxed{\bold{ \frac{1}{ \tan( \alpha ) } = \cot( \alpha ) }} \\ \\ \star \boxed{ \bold{ \tan( \frac{\pi}{2} - \alpha ) = \cot \alpha }} \\ \\ \star \boxed{ \bold{ { \cot}^{ - 1} ( \cot \theta) = \theta}}$

Answer 2

Step-by-step explanation:

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