Answer
See the explanation below.
Work Step by Step
$\frac{{{\cos }^{2}}t+4\cos t+4}{\cos t+2}$
Factorize the numerator of the above expression as:
$\begin{align}
& \frac{{{\cos }^{2}}t+4\cos t+4}{\cos t+2}=\frac{{{\cos }^{2}}t+2\cos t+2\cos t+4}{\cos t+2} \\
& =\frac{\cos t\left( \cos t+2 \right)+2\left( \cos t+2 \right)}{\cos t+2} \\
& =\frac{\left( \cos t+2 \right)\left( \cos t+2 \right)}{\cos t+2} \\
& =\cos t+2
\end{align}$
Now, consider the right side of the provided expression:
$\frac{2\sec t+1}{\sec t}$
Factorize the above expression as:
$\frac{2\sec t+1}{\sec t}=\frac{2\sec t}{\sec t}+\frac{1}{\sec t}$
By using the reciprocal identity $\cos t=\frac{1}{\sec t}$ , then the above expression can be further simplified as:
$\begin{align}
& \frac{2\sec t}{\sec t}+\frac{1}{\sec t}=2+\cos t \\
& =\cos t+2
\end{align}$
Now, the identity is verified because both sides are equal to $\cos t+2$.
Hence, the left side of the expression is equal to the right side, which is
$\frac{{{\cos }^{2}}t+4\cos t+4}{\cos t+2}=\frac{2\sec t+1}{\sec t}$.
