Question

Verify the identity.

cotangent of x to the second power divided by quantity cosecant of x plus one equals quantity one minus sine of x divided by sine of x

Answer 1

I don't know what to do

Answer 2

I'd leave the right side alone and see whether I can get the left side to equal the right side (that is, I'd try to prove the identity).

Answer 3

I'm assuming I understood your question and have correctly interpreted it to look like the following:

Answer 4

\[\frac{\cot^2{x}}{\csc{x}+1} = \frac{1-\sin{x}}{sinx}\]

Answer 5

\[\frac{ (\cot x)^{2} }{ (\csc x)+1 }=\frac{ \frac{ (\cos x)^{2} }{ (\sin x)^{2} } }{ \frac{ 1 }{ \sin x }+\frac{ \sin x }{ \sin x }}\]

Answer 6

Are you interpreting the right side the same as me, @mathmale?

Answer 7

@Indivicivet : yes, indeed. I'm leaving the right side alone and am in the process of re-writing the left side in the hope that I can get it to equal the right side.

Answer 8

MLS: thanks for posting that illustration!!! Whenever you have an illustration, please share it.

Answer 9

Sorry, I'm still new at working this

Answer 10

@mathmale @MoonLightSkanada in my opinion writing it out using the equation editor (provided it's fairly simple and thus not too much work!) is preferable since then it shows up as more than a thumbnail.

Answer 11

Anyway I'll let @mathmale handle this one!

Answer 12

Thanks for your courtesy, Indivicivet!

Answer 13

Going back to my most recent result, the next step produces the following:

Answer 14

\[\frac{ \frac{ 1-\sin ^{2}x }{ \sin ^{2} x} }{ \frac{ 1+\sin x }{ \sin x} }=\frac{ 1-\sin ^{2}x }{ (\sin x)(1+\sin x) }\]

Answer 15

MLS: would you please factor 1-(sin x)^2? Once you've done that, reduce the main fraction. Looking for ward to seeing what you find.

Answer 16

\[1-\sin ^{2} =\cos ^{2}\] so that would be \[\frac{ \cos ^{2} }{ \sin ^{2}} ?\]

Answer 17

MLS: Your response is an identity, not a case of factoring.

Hint:
\[a ^{2}-b ^{2}=(a-b)(a+b).\]This is factoring. Apply this procedure to 1-(sin x)^2.

Answer 18

So that would be 1+sinx an 1-sinx

Answer 19

Perfect. Now, ask yourself: Does the left side of your equation equal the right side?

Answer 20

Yes

Answer 21

Yay!

Answer 22

COOL! See you again?