Verify the formula using any method.
![\[ \int \frac{\cos^m x}{\sin^n x} \, dx = -\frac{\cos^{m+1} x}{(n-1)\sin^{n-1} x} - \frac{m-n+2}{n-1} \int \frac{\cos^m x}{\sin^{n-2} x} \, dx + C \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-ff49bce8576130fb4862a4439d6e569f_l3.png "Rendered by QuickLaTeX.com")
for 
.
Proof. As a first step, we write
![\[ \int \frac{\cos^m x}{\sin^n x} \, dx = \int \frac{\cos^{m+2} x}{\sin^n x} \sec^2 x \, dx. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-9e42540cf7a3a7c86557bae301d36771_l3.png "Rendered by QuickLaTeX.com")
Then, we integrate by parts with
![\begin{align*} u &= \frac{\cos^{m+2} x}{\sin^n x} & du &= \frac{-(m+2) \cos^{m+1} x \sin^{n+1} x - n \sin^{n-1} x \cos^{m+3} x}{\sin^{2n} x} \, dx \\[9pt] &&&= \frac{-(m+2) \cos^{m+1} x \sin^2 x - n \cos^{m+3} x}{\sin^{n+1} x} \, dx \\[9pt] &&&= -(m+2) \frac{\cos^{m+1} x}{\sin^{n-1} x} - n \frac{\cos^{m+3} x}{\sin^{n+1} x} \, dx\\[9pt] dv &= \sec^2 x \, dx & v &= \tan x. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-5af8a62321be5f14d8ca994393421dde_l3.png "Rendered by QuickLaTeX.com")
This gives us
![\begin{align*} \int \frac{\cos^m x}{\sin^n x} &= \int \frac{\cos^{m+2} x}{\sin^n x} \sec^2 x \, dx \\[9pt] &= uv - \int v \, du \\[9pt] &= \frac{\cos^{m+1} x}{\sin^{n-1} x} - \int \left( -(m+2) \frac{\cos^m x}{\sin^{n-2} x} - n \frac{\cos^{m+2} x}{\sin^n x} \right) \, dx \\[9pt] &= \frac{\cos^{m+1} x}{\sin^{n-1} x} + (m+2) \int \frac{\cos^m x}{\sin^{n-2} x} \, dx + n \int \frac{\cos^{m+2} x}{\sin^n x} \, dx \\[9pt] &= \frac{\cos^{m+1} x}{\sin^{n-1} x} + (m+2) \int \frac{\cos^m x}{\sin^{n-2} x} \, dx + n \int \frac{\cos^m x (1 - \sin^2 x)}{\sin^n x} \, dx \\[9pt] &= \frac{\cos^{m+1} x}{\sin^{n-1} x} + (m+2) \int \frac{\cos^m x}{\sin^{n-2} x} \, dx + n \int \frac{\cos^m x}{\sin^n x} \, dx - n \int \frac{\cos^m x}{\sin^{n-2} x} \, dx. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-d7b27677fd85e2a835eaf85d3d6bbda3_l3.png "Rendered by QuickLaTeX.com")
Moving the 
back to the left we then have,
![\begin{align*} (1-n) \int \frac{\cos^m x}{\sin^n x} \, dx &= \frac{\cos^{m+1} x}{\sin^{n-1} x} + (m+2-n) \int \frac{\cos^m x}{\sin^{n-2} x} \, dx \\[9pt] \implies \qquad \int \frac{\cos^m x}{\sin^n x} \, dx &= -\frac{\cos^{m+1} x}{(n-1)\sin^{n-1} x} - \frac{m-n+2}{n-1} \int \frac{\cos^m x}{\sin^{n-2} x} \, dx. \qquad \blacksquare \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a92b75c1bdfd2da2fa2fc662800e767d_l3.png "Rendered by QuickLaTeX.com")
Verify the formula using any method.
for .
Proof. As a first step, we write
Then, we integrate by parts with
This gives us
Moving the back to the left we then have,