Question

The number of roots of the equation $sin^{- 1}x-cos^{- 1}x=sin^{- 1}\left(5 x - 3\right)$ is/are

• A. $3$
• B. $1$
• C. $2$
• D. $0$

Answer 1

Answer: (B)

$sin^{- 1}x-cos^{- 1}x=sin^{- 1}\left(5 x - 3\right)$
$\Rightarrow \frac{\pi }{2}-cos^{- 1}x-cos^{- 1}x=\frac{\pi }{2}-cos^{- 1}\left(5 x - 3\right)$
$\Rightarrow 2cos^{- 1}x=cos^{- 1}\left(5 x - 3\right). \, Also \, x\in \left[- 1,1\right]$ … $\left(1\right)$
$\Rightarrow cos^{- 1}\left(2 x^{2} - 1\right)=cos^{- 1}\left(5 x - 3\right)$ and $\left(5 x - 3\right)\in \left[- 1,1\right],$ i.e., $-1\leq 5x-3\leq 1$
$\Rightarrow 2x^{2}-1=5x-3, \, hence, \, x\in \left[\frac{2}{5} , \frac{4}{5}\right]$
$\Rightarrow 2x^{2}-5x+2=0\Rightarrow x=2$ or $\frac{1}{2}$
but, $x=2$ does not satisfy the equation $\left(1\right)$
Hence, the given equation has only one root