Question

answer this with the formulas​

Answer 1

${\huge{\bf{\red{\underline{Solution:}}}}}$

${\bf{\blue{\underline{Given:}}}}$

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

${\sf{\underline{\blue{Now,}}}}$

$\star \boxed {\sf{\orange{ put \: x = sec \theta}}}$

Then,

${\tt{ \implies \: \sqrt{ {x}^{2} - 1 } }}$

${\tt{ \implies \sqrt{ {sec}^{2} \theta - 1 } }}$

${\tt{ \implies tan \theta }}$

${\bf{\blue{\underline{Therefore,}}}}$

${\tt{ {tan}^{ - 1} \frac{1}{ \sqrt{ {x}^{2} - 1} } = {tan}^{ - 1} \bigg( \frac{1}{tan} \bigg) }}$

${\tt{ \implies {tan}^{ - 1}(cot \theta) }}$

${\tt{ \implies {tan}^{ - 1} \bigg( tan\big( \frac{\pi}{2} - \theta \big) \bigg)}}$

${\tt{ \implies \frac{\pi}{2} - \theta }}$

${\bf{\blue{\underline{Hence,}}}}$

$\boxed {\tt{\purple{ \bigstar \: \: {tan}^{ - 1} \frac{1}{ \sqrt{ {x}^{2} - 1} } = \frac{\pi}{2} - {sec}^{ - 1} x}}}$