First slide

Properties of ITF

Question

$C o s^{2} [T a n^{- 1} (S i n (C o t^{- 1} x))] =$

Moderate

A

$\frac{x^{2} + 1}{x^{2} + 2}$
B

$\frac{x^{2} - 1}{x^{2} + 2}$
C

$\frac{x^{2} + 1}{x^{2} - 2}$
D

$\frac{x^{2} - 1}{x^{2} + 1}$

Solution

$W e h a v e$
$C o t^{- 1} x = S i n^{- 1} (\frac{1}{\sqrt{1 + x^{2}}})$ and $T a n^{- 1} x = C o s^{- 1} (\frac{1}{\sqrt{1 + x^{2}}})$
$∴ \cos^{2} [\tan^{- 1} (\sin (c o t^{- 1} x))] = \cos^{2} [\tan^{- 1} (\frac{1}{\sqrt{1 + x^{2}}})] = {[\frac{1}{\sqrt{1 + {(\frac{1}{\sqrt{1 + x^{2}}})}^{2}}}]}^{2} = \frac{x^{2} + 1}{x^{2} + 2}$

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