# How do you prove (1+sinx)/(1-sinx)+(sinx-1)/(1+sinx)?

Oct 23, 2016

The expression simplifies to $4 \sin x {\sec}^{2} x$.

#### Explanation:

Put on a common denominator.

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

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

Apply the identity ${\sin}^{2} \theta + {\cos}^{2} \theta = 1 \to {\cos}^{2} \theta = 1 - {\sin}^{2} \theta$ to the numerator and simply the denominator.

$= \frac{4 \sin x}{\cos} ^ 2 x$

Apply the identity $\frac{1}{\cos} \beta = \sec \beta$.

$= 4 \sin x {\sec}^{2} x$

Hopefully this helps!