If ( ^ - 1 (x + 1) = ( ^ - 1 x), then x = - Vidyadip

MCQ

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

✓
$ - \frac{1}{2}$
B
$\frac{1}{2}$
C
$0$
D
$\frac{9}{4}$

✓

Answer

Correct option: A.

$ - \frac{1}{2}$

a
(a) $\sin [{\cot ^{ - 1}}(x + 1)] = \sin \left( {{{\sin }^{ - 1}}\frac{1}{{\sqrt {{x^2} + 2x + 2} }}} \right)$

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

$\cos ({\tan ^{ - 1}}x) = \cos \left( {{{\cos }^{ - 1}}\frac{1}{{\sqrt {1 + {x^2}} }}} \right) = \frac{1}{{\sqrt {1 + {x^2}} }}$

Thus, $\frac{1}{{\sqrt {{x^2} + 2x + 2} }} = \frac{1}{{\sqrt {1 + {x^2}} }}$

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

$ \Rightarrow $ $x = - \frac{1}{2}$.

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