QJEEM17Mace4c: JEE Main 2017 Mathematics - Class 12 JEEM17Mace4c Solution: A value of x satisfying the equation sin[cot −1 (1+ x)] = cos…

Let, ${\cot ^{ - 1}}(1 + x) = \alpha$

$\Rightarrow 1 + x = \cot \alpha$

Now, let ${\tan ^{ - 1}}(x) = \beta$

$\Rightarrow x = \tan \beta$

Given, $\sin ({\cot ^{ - 1}}(x)) = \cos ({\tan ^{ - 1}}(x))$

$\Rightarrow \sin \alpha = \cos \beta$

From $\Delta$ ABC, $\sin \alpha = {1 \over {\sqrt {1 + {x^2} + 2x + 1} }}$

$= {1 \over {\sqrt {{x^2} + 2x + 2} }}$

From $\Delta$ MNO, $\cos \beta = {1 \over {\sqrt {{x^2} + 1} }}$

$\therefore$ ${1 \over {\sqrt {{x^2} + 2x + 2} }} = {1 \over {\sqrt {{x^2} + 1} }}$

$\Rightarrow {x^2} + 2x + 2 = {x^2} + 1$

$\Rightarrow 2x + 2 = 1$

$\Rightarrow 2x = - 1$

$\Rightarrow x = -{1 \over 2}$