# How do you simplify csc X + cot X = sin X / (1+cos X)?

Mar 22, 2016

This is not a valid identity.

#### Explanation:

We can prove this is invalid by using a test value of $x = \frac{\pi}{4}$:

$\csc \left(\frac{\pi}{4}\right) + \cot \left(\frac{\pi}{4}\right) \ne \sin \frac{\frac{\pi}{4}}{1 + \cos \left(\frac{\pi}{4}\right)}$

$\sqrt{2} + 1 \ne \frac{\frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}}$

$\sqrt{2} + 1 \ne \frac{1}{\sqrt{2} \left(1 + \frac{1}{\sqrt{2}}\right)}$

$\sqrt{2} + 1 \ne \frac{1}{\sqrt{2} + 1}$

In fact, as we can might see is happening here, these functions are actually reciprocals of one another: they only intersect when their values equal $1$ or $- 1$.

We can also prove these are not equal by attempting to simplify the functions:

$\csc x + \cot x \ne \sin \frac{x}{1 + \cos x}$

$\frac{1}{\sin} x + \cos \frac{x}{\sin} x \ne \sin \frac{x}{1 + \cos x}$

$\frac{1 + \cos x}{\sin} x \ne \sin \frac{x}{1 + \cos x}$

Indeed, these functions are reciprocals of one another so the identity is invalid.