# How do you verify (1/(sinx+1))+(1/(cscx+1))=1?

Apr 16, 2015

We know that color(blue)(csc x = 1/sinx

Therefore
$\left(\frac{1}{\sin x + 1}\right) + \left(\frac{1}{\csc x + 1}\right)$

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

Multiplying the Numerator and the Denominator of the second term with $\sin x$, we get

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

As the Denominators are the same, we can simply add the numerators over the common denominator

$= \frac{\cancel{1 + \sin x}}{\cancel{1 + \sin x}}$

$= 1$ (which is the Right Hand Side)

Hence Proved.