Question

For the equation $ \cos^{-1}x + \cos^{-1}2x + 2\pi =0 $, the number of real solution(s) is

Answer 1

$ \cos^{-1}x + \cos^{-1}2x = - 2\pi $
$\Rightarrow \cos^{-1}2x= -2\pi -\cos^{-1}x $
$\Rightarrow 2x=\cos( 2\pi + \cos^{-1}x )$
$\Rightarrow 2x=\cos (\cos^{-1}x)$
$\Rightarrow 2x=x$
$\Rightarrow x=0 $
$\Rightarrow$ But $x=0 $ does not satisfy the given equation.
Hence, no real solution.

Alternate solution :
All of the terms $ \cos^{-1}x, \cos^{-1}2x $ and $ 2\pi $ are positive or non-negative. So, the sum of $3$ positive terms can't be equal to zero. Hence no real solution.