Question

Verify the identity.

Answer 1

Identity to verify:

$\dfrac{\cot x}{1+\csc x}=\dfrac{\csc x-1}{\cot x}$

Recall that

$\cos^2x+\sin^2x=1$

Divide both sides by $\sin^2x$ and we get

$\cot^2x+1=\csc^2x$

or

$\cot^2x=\csc^2x-1=(\csc x-1)(\csc x+1)$

So if we multiply the numerator and denominator of

$\dfrac{\cot x}{1+\csc x}$

by $\csc x-1$, we get

$\dfrac{\cot x(\csc x-1)}{(1+\csc x)(\csc x-1)}=\dfrac{\cot x(\csc x-1)}{\csc^2x-1}=\dfrac{\cot x(\csc x-1)}{\cot^2x}$

Then as long as $\cot x\neq0$, we can cancel terms to end up with

$\dfrac{\csc x-1}{\cot x}$

and establish the identity.