Question

The number of solutions of the equation $cot xcos⁡x-1=cot⁡x-cos⁡x, \, \forall x\in \left[0 , 2 \pi \right]$ is equal to

Answer 1

$cot x \cdot cos x - cot x + \left(cos x - 1\right) = 0$
$\Rightarrow \left(cot x + 1\right)\left(cos ⁡ x - 1\right)=0$
$\Rightarrow cot x=-1$ or $cos x=1$ (Not possible as $sin x\neq 0$ )
Hence, $2$ solutions are possible in $\left[0 , 2 \pi \right]$