Answer
$\dfrac{2+5\cos{x}}{\sin{x}}=2\csc{x}+5\cot{x}$ is an identity
Work Step by Step
Use a graphing utility to graph $y=\dfrac{2+5\cos{x}}{\sin{x}}$ and $y=2\csc{x}+5\cot{x}$.
(Refer to the graphs below.)
Note that the graphs are exactly the same as all the curves coincide.
This means that the given equation is an identity.
RECALL:
(1) $\csc{x}=\dfrac{1}{\sin{x}}$
(2) $\cot{x}=\dfrac{\cos{x}}{\sin{x}}$
Use the definitions above to obtain:
\begin{align*}
\dfrac{2+5\cos{x}}{\sin{x}}&=\frac{2}{\sin{x}}+\frac{5\cos{x}}{\sin{x}}\\\\
&=2\left(\frac{1}{\sin{x}}\right)+5\left(\frac{\cos{x}}{\sin{x}}\right)\\\\
&=2\csc{x}+5\cot{x}
\end{align*}
Therefore,
$$\dfrac{2+5\cos{x}}{\sin{x}}=2\csc{x}+5\cot{x}$$
