Question

The number of integral values of k for which the equation ${\sin }^{-1}x+{\tan }^{-1}x=2k+1$ has a solutions is:

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

Given,

${\sin }^{-1}x+{\tan }^{-1}x=2k+1$

Let,

$y_1={\sin }^{-1}x+{\tan }^{-1}x$

$y_2=2k+1$

$y_1={\sin }^{-1}x+{\tan }^{-1}x$ is continuous and differentiable function with

Domain $=-1\le x\le 1$

Range $(-\frac{\pi }{2}-\frac{\pi }{4},\frac{\pi }{2}+\frac{\pi }{2})=(-\frac{3\pi }{4},\frac{3\pi }{4})\approx (-2.356,2.356)$

$y_2=2k+1$

$=-3,-1,1,3,5,........$

$y_1,y_2$ intersect at $-1,1$ only

$2k+1=1\Rightarrow k=0$

$2k+1=-1\Rightarrow k=-1$