Question

Verify the identity.
cotx / 1+cscx= cscx-1 / cotx

Answer 1

There are a bunch of handy identities which involve the difference or sum of squares.\[\cos^2 x = 1 - \sin^2x, \qquad \sec^2x = 1+\tan^2x, \qquad \csc^2x=1+\cot^2x\] We can think of the 1 as 1^2 in these problems.
You really want to remember how to manipulate the difference of 2 squares, because it comes up over and over and over. :)

So here we'll want to take advance of that.
Let's leave the RIGHT side alone, and try to make make the left side match it.

\[\huge a^2-b^2=(a+b)(a-b)\]Remember this identity? :D The Difference of Squares can be broken down into CONJUGATES like this.

Answer 2

Ok

Answer 3

I know that cotx= csc^2x -1 @zepdrix

Answer 4

Please help me I have a timed test D:

Answer 5

@zepdrix

Answer 6

cot^2 x u mean? ^^

Answer 7

Ok if you are familiar with that identity, then let's manipulate the RIGHT instead, it'll be easier to work with :)

Answer 8

\[\huge \frac{\csc x - 1}{\cot x}\left(\frac{\csc x + 1}{\csc x + 1}\right)\]We'll multiply the top and bottom by this fraction, giving us,\[\huge \frac{\csc^2x-1}{\cot x(\csc x + 1)}\]

Answer 9

We don't want to multiply out the bottom!
Because as you might see, the top we can now apply that nice identity to!
And we'll have a nice cancellation after that.

Answer 10

But cotx is on the bottom...

Answer 11

Can you please tell me the answer?

Answer 12

"I know that cot^2x= csc^2x -1"
You said this earlier! :D

See how we have a csc^2x-1 on top?
Replace it with cot^2x. c:

Answer 13

OH

Answer 14

You rock