How do you simplify sinx/(1 + sinx) - sinx/(1 -sinx)?

Apr 17, 2016

You should get $- 2 {\tan}^{2} x$ as a final answer.

Explanation:

$\frac{\sin x}{1 + \sin x} - \sin \frac{x}{1 - \sin x}$

=(sinx(1 - sinx) - sinx(1 + sinx))/((1 + sinx)(1 - sinx)

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

Using the pythagorean identity ${\cos}^{2} x + {\sin}^{2} x = 1$

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

Using the quotient identity $\tan x = \sin \frac{x}{\cos} x$

$= - 2 {\tan}^{2} x$

Hopefully this helps!