Answer
The solution of the given question is $2\csc x$.
Work Step by Step
$\frac{1+\cos x}{\sin x}+\frac{\sin x}{1+\cos x}$
Then, use a common denominator for the given equation:
$\begin{align}
& \frac{1+\cos x}{\sin x}+\frac{\sin x}{1+\cos x}=\frac{1+\cos }{\sin x}\times \frac{1+\cos x}{1+\cos x}+\frac{\sin x}{1+\cos x}\times \frac{\sin x}{\sin x} \\
& =\frac{1+2\cos x+{{\cos }^{2}}x}{\left( \sin x \right)\left( 1+\cos x \right)}+\frac{{{\sin }^{2}}x}{\left( \sin x \right)\left( 1+\cos x \right)}
\end{align}$
The two fractions will be simplified by creating a common base as shown below:
$\begin{align}
& \frac{1+2\cos x+{{\cos }^{2}}x}{\left( \sin x \right)\left( 1+\cos x \right)}+\frac{{{\sin }^{2}}x}{\left( \sin x \right)\left( 1+\cos x \right)}=\frac{1+2\cos x+{{\cos }^{2}}x+{{\sin }^{2}}x}{\left( \sin x \right)\left( 1+\cos x \right)} \\
& =\frac{1+2\cos x+1}{\left( \sin x \right)\left( 1+\cos x \right)} \\
& =\frac{2+2\cos x}{\left( \sin x \right)\left( 1+\cos x \right)} \\
& =\frac{2\left( 1+\cos x \right)}{\left( \sin x \right)\left( 1+\cos x \right)}
\end{align}$
By using the reciprocal identity $\frac{1}{\sin x}=\csc x$, we get:
$\begin{align}
& \frac{2\left( 1+\cos x \right)}{\left( \sin x \right)\left( 1+\cos x \right)}=\frac{2}{\sin x} \\
& =\frac{2}{\sin x} \\
& =2\csc x
\end{align}$
Conjecture: Left side is equal to $2\csc x$.
Thus, the left side of the expression is equal to the right side, which is $\frac{1+\cos x}{\sin x}+\frac{\sin x}{1+\cos x}=2\csc x$.
