Answer
The solution of the given question is $2\sin x$.
Work Step by Step
$\frac{\cos x+\cot x\sin x}{\cot x}$
We divide $\cos x$ and $\cot x\sin x$ in the numerator separately by $\cot x$
$\begin{align}
& \frac{\cos x+\cot x\sin x}{\cot x}=\frac{\cos x}{\cot x}+\frac{\cot x\sin x}{\cot x} \\
& =\frac{\cos x}{\frac{\cos x}{\sin x}}+\frac{\cot x\sin x}{\cot x} \\
& =\frac{\cos x\sin x}{\cos x}+\sin x
\end{align}$
Further solving,
$\begin{align}
& \frac{\cos x\sin x}{\cos x}+\sin x=\sin x+\sin x \\
& =2\sin x
\end{align}$
Conjecture: Left side is equal to $2\sin x$.
Thus, the left side of the expression is equal to the right side, which is $\frac{\cos x+\cot x\sin x}{\cot x}=2\sin x$.
