# How do you prove the identity (1-cos2x) / tanx = sin2x?

Oct 1, 2015

See the explanation below.

#### Explanation:

Use one of the identities:

$\cos 2 x = {\cos}^{2} x - {\sin}^{2} x$
$\cos 2 x = 1 - 2 {\sin}^{2} x$
$\cos 2 x = 2 {\cos}^{2} x - 1$

Playing around with them on scratch paper (or thinking about them) will lead to using the second version.

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

$= \frac{2 {\sin}^{2} x}{\sin \frac{x}{\cos} x}$

$= 2 {\sin}^{2} x \cdot \cos \frac{x}{\sin} x$

$= 2 \sin x c o n s x$

$= \sin 2 x$