Verify the identity. cotangent of x divided by quantity one plus cosecant of x equals quantity cosecant of x minus one divided by cotangent of x

Question

asked Jun 7, 2019 211k views

1 Answer

NicholasByDesign · Answer 1 · 2019-06-12T05:03:56+0000

Identity to verify:

$(\cot x)/(1+\csc x)=(\csc x-1)/(\cot x)$

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

$(\cot x)/(1+\csc x)$

by
$\csc x-1$ , we get

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

Then as long as
$\cot x\\eq0$ , we can cancel terms to end up with

$(\csc x-1)/(\cot x)$

and establish the identity.