# How do you prove  [1 - cos(x)]/[sin(x)] = tan(x/2)?

Oct 1, 2016

#### Explanation:

As $\cos x = 1 - 2 {\sin}^{2} \left(\frac{x}{2}\right)$ and $\sin x = 2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)$

Hence $\frac{1 - \cos x}{\sin} x$

= $\frac{1 - \left(1 - 2 {\sin}^{2} \left(\frac{x}{2}\right)\right)}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}$

= $\frac{1 - 1 + 2 {\sin}^{2} \left(\frac{x}{2}\right)}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}$

= $\frac{2 {\sin}^{2} \left(\frac{x}{2}\right)}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}$

= $\frac{2 \sin \left(\frac{x}{2}\right) \sin \left(\frac{x}{2}\right)}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}$

= $\sin \frac{\frac{x}{2}}{\cos} \left(\frac{x}{2}\right)$

= $\tan \left(\frac{x}{2}\right)$

Oct 1, 2016

#### Explanation:

$\sin \left(\frac{x}{2} + \frac{x}{2}\right)$

$= \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) + \cos \left(\frac{x}{2}\right) \sin \left(\frac{x}{2}\right)$

$= \sin \left(\frac{x}{2}\right) \left\{\cos \left(\frac{x}{2}\right) + \cos \left(\frac{x}{2}\right)\right\}$

$= \sin \left(\frac{x}{2}\right) \cdot 2 \cos \left(\frac{x}{2}\right)$

$= 2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)$

$= \sin \left(x\right)$

This is because:

$\sin \left(A + B\right) = \sin \left(A\right) \cos \left(B\right) + \cos \left(A\right) \sin \left(B\right)$

$\cos \left(\frac{x}{2} + \frac{x}{2}\right)$

$= \cos \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) - \sin \left(\frac{x}{2}\right) \sin \left(\frac{x}{2}\right)$

$= {\cos}^{2} \left(\frac{x}{2}\right) - {\sin}^{2} \left(\frac{x}{2}\right)$

$= 1 - {\sin}^{2} \left(\frac{x}{2}\right) - {\sin}^{2} \left(\frac{x}{2}\right)$

$= 1 - 2 {\sin}^{2} \left(\frac{x}{2}\right)$

$= \cos \left(x\right)$

This is because:

$\cos \left(A + B\right) = \cos \left(A\right) \cos \left(B\right) - \sin \left(A\right) \sin \left(B\right)$

Which means that:

$L H S$

$= \frac{1 - \cos \left(x\right)}{\sin} \left(x\right)$

$= \frac{1 - \left\{1 - 2 {\sin}^{2} \left(\frac{x}{2}\right)\right\}}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}$

$= \frac{1 - 1 + 2 {\sin}^{2} \left(\frac{x}{2}\right)}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}$

$= \frac{2 {\sin}^{2} \left(\frac{x}{2}\right)}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}$

$= \frac{2 \sin \left(\frac{x}{2}\right) \cdot \sin \left(\frac{x}{2}\right)}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}$

$= \frac{\sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}$

$= \tan \left(\frac{x}{2}\right)$

$= R H S$