Question

If $\sin \left[{\cot }^{-1}\right(x+1\left)\right]=\cos \left({\tan }^{-1}x\right)$, then find $x$.

Answer 1

$\sin$ and $\cos$ can be equal in 1st and 3rd quadrants which is $\frac{\pi }{4}$ and $\frac{5\pi }{4}$.

Considering $\frac{\pi }{4}$ as solution we get $x=0$ and $1$ from LHS and RHS respectively which can't be a solution.

Similarly considering $\frac{5\pi }{4}$we get $x=0$ and $1$ from LHS and RHS respectively.

Hence, no values of $x$ satisfies the equation.