To solve the integral \(\int \frac{\sin^n x}{\cos^{n+2} x} \, dx\), we can simplify it by using a substitution method involving trigonometric identities.
First, recall the identity \(\tan x = \frac{\sin x}{\cos x}\). Therefore:
\[\frac{\sin^n x}{\cos^{n+2} x} = \frac{\sin^n x}{\cos^n x \cdot \cos^2 x} = \tan^n x \cdot \sec^2 x\]
Now, our integral becomes:
\[\int \tan^n x \cdot \sec^2 x \, dx\]
For this transformed integral, apply the substitution \(u = \tan x\) which gives \(\frac{du}{dx} = \sec^2 x\). Thus, \(du = \sec^2 x \, dx\).
Substitute to get:
\[\int u^n \, du\]
This is a straightforward power integral. Using the formula for integration of powers, we have:
\[\int u^n \, du = \frac{u^{n+1}}{n+1} + C\]
Substitute back \(u = \tan x\) to express the answer in terms of x:
\[\frac{\tan^{n+1} x}{n+1} + C\]
Thus, the solution to the integral \(\int \frac{\sin^n x}{\cos^{n+2} x} \, dx\) is \(\frac{\tan^{n+1} x}{n+1} + C\).
Therefore, the correct option is: