Question

Prove that

$sec {}^{ - 1} x + {cosec}^{ - 1} x = \frac{\pi}{2}$
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Answer 1

$\bf \: Let \: \: \: \: {sec}^{ - 1} x = \theta \: \: \: \: \: \: \: \: \: \: \: \: \: .......(1)\\ \\ so \: \: \bf \: \: \: x = \: sec \: \theta \:$

$\bf \: We \: know \: that \: Formula \: \boxed{ \bf \: cosec ( \frac{\pi}{2} - \theta )= sec \: \theta}$

Now, putting it value

$\bf \: \: cosec \: ( \frac{\pi}{2} - \theta) = x \\ \\ \\ \implies \: \bf \: \: ( \frac{\pi}{2} - \theta) = {cosec}^{ - 1} x \\ \\ \: lets \: put \: the \: value \: of \: \theta \: from \: the \: first \: equation \\ \\ \\ \implies \: \bf \: \: ( \frac{\pi}{2} - {sec}^{ - 1} x) = {cosec}^{ - 1} x \\ \\ \\ \implies \: \boxed{\bf \: \: sec {}^{ - 1}x + cosec {}^{ - 1} x = \frac{\pi}{2} }$