Verify the formula using any method.
![\[ \int \frac{\cos^m x}{\sin^n x} \, dx = \frac{\cos^{m-1} x}{(m-n)\sin^{n-1} x} + \frac{m-1}{m-n} \int \frac{\cos^{m-2} x}{\sin^n x} \, dx + C \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-f706b24b0ae9015e493bda500bfdb632_l3.png "Rendered by QuickLaTeX.com")
for 
.
Proof. First, we recall from a previous exercise (Section 5.10, Exercise #19) that we used integration by parts to establish the following formula,
![\[ \int \frac{\cos^{m+1} x}{\sin^{n+1} x} \, dx = -\frac{1}{n} \cdot \frac{\cos^m x}{\sin^n x} - \frac{m}{n} \int \frac{\cos^{m-1} x}{\sin^{n-1} x} \, dx. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-655b452b824e39e7d006385f66c1029a_l3.png "Rendered by QuickLaTeX.com")
Applying this to the current integral (with 
in place of 
and 
in place of 
) and computing, we have
![\begin{align*} \int \frac{\cos^m x}{\sin^n x} \, dx &= -\frac{1}{n-1} \cdot \frac{\cos^{m-1} x}{\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2} x}{\sin^{n-2} x} \, dx \\[9pt] &= -\frac{1}{n-1} \cdot \frac{\cos^{m-1} x}{\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2} x \sin^2 x}{\sin^n x} \, dx \\[9pt] &= -\frac{1}{n-1} \cdot \frac{\cos^{m-1} x}{\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2}x (1 - \cos^2 x)}{\sin^n x} \, dx \\[9pt] &= -\frac{\cos^{m-1} x}{(n-1)\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2}x}{\sin^n x} \, dx + \frac{m-1}{n-1} \int \frac{\cos^m x}{\sin^n x} \, dx. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-d1fcd2239faaefb489212e83eb756e34_l3.png "Rendered by QuickLaTeX.com")
Bringing the last integral 
back to the left, we have
![\begin{align*} && \left( 1 - \frac{m-1}{n-1} \right) \int \frac{\cos^m x}{\sin^n x} \, dx &= -\frac{\cos^{m-1} x}{(n-1)\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2} x}{\sin^n x} \, dx \\[9pt] \implies && \frac{n-m}{n-1} \int \frac{\cos^m x}{\sin^n x} \, dx &= -\frac{\cos^{m-1} x}{(n-1)\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2} x}{\sin^n x} \, dx \\[9pt] \implies && \int \frac{\cos^m x}{\sin^n x} \, dx &= -\frac{\cos^{m-1} x}{(n-m)\sin^{n-1} x} - \frac{m-1}{n-m} \int \frac{\cos^{m-2} x}{\sin^n x} \, dx \\[9pt] \implies && \int \frac{\cos^m x}{\sin^n x} \, dx &= \frac{\cos^{m-1} x}{(m-n) \sin^{n-1} x} + \frac{m-1}{m-n} \int \frac{\cos^{m-2} x}{\sin^n x} \, dx. \qquad \blacksquare \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-5633f751d9b72c4f0012097c315c53b3_l3.png "Rendered by QuickLaTeX.com")