Evaluate the following integrals:
$\int \frac{1-\cos x}{1+\cos x} d x$
Let $I=\int \frac{1-\cos x}{1+\cos x} d x$
We have $\cos x=\cos \left(2 \times \frac{x}{2}\right)$
We know $\cos 2 \theta=1-2 \sin ^{2} \theta=2 \cos ^{2} \theta-1$
Hence, in the numerator, we can write $1-\cos x=2 \sin ^{2} \frac{x}{2}$
In the denominator, we can write $1+\cos x=2 \cos ^{2} \frac{x}{2}$
Therefore, we can write the integral as
$I=\int \frac{2 \sin ^{2} \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}} d x$
$\Rightarrow I=\int \frac{\sin ^{2} \frac{x}{2}}{\cos ^{2} \frac{x}{2}} d x$
$\Rightarrow I=\int \tan ^{2} \frac{x}{2} d x$
$\Rightarrow \mathrm{I}=\int\left(\sec ^{2} \frac{\mathrm{x}}{2}-1\right) \mathrm{dx}\left[\because \sec ^{2} \theta-\tan ^{2} \theta=1\right]$
$\Rightarrow I=\int \sec ^{2} \frac{x}{2} d x-\int d x$
Recall $\int \sec ^{2} x d x=\tan x+c$ and $\int d x=x+c$
$\Rightarrow \mathrm{I}=\frac{\tan \frac{\mathrm{x}}{2}}{\frac{1}{2}}-\mathrm{x}+\mathrm{c}$
$\therefore I=2 \tan \frac{x}{2}-x+c$
Thus, $\int \frac{1-\cos x}{1+\cos x} d x=2 \tan \frac{x}{2}-x+c$
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