Exhaustive set of values of parameter 'a' so that $sin^{-1} x - tan^{-1}x= a$ has a solution

$\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$

Let $f(x) = sin^{-1}x -tan^{-1}x.$

Clearly, f(x) is defined for all x ∈ [-1,1].

Also,

$f'(x)=\frac{1}{\sqrt{1-x^2}}-\frac{1}{1+x^2} > 0 $ for all x ∈ (-1, 1)

$⇒f(-1) ≤f(x)≤f(1)$

$⇒ -\frac{\pi}{2}+\frac{\pi}{4} < sin^{-1} x - tan^{-1}x ≤ \frac{\pi}{2}-\frac{\pi}{4}$

$⇒ -\frac{\pi}{4}≤ a ≤ \frac{\pi}{4} ⇒ a ∈ \left[-\frac{\pi}{4},\frac{\pi}{4}\right]$