Prove (cos^2x-sin^2x+sin^4x)/(sin^4x) = cot^4x

Question

anonymous · Answer

Trying to solve csc^4x-2csc^2x+1=cot^4x...this is as far as I got.

anonymous · Answer

is it...$$\frac{ \cos ^{2}x-\sin ^{2}x+\sin ^{4}x }{ \sin ^{4}x }=\cot ^{4}x$$

anonymous · Answer

Yes...unless I made a mistake along the way.

anonymous · Answer

we can start factoring sin^2x...

anonymous · Answer

all expression of (1-sin^2x) = cos^2x....

anonymous · Answer

oops i can't finish it... i have to go, but if you'll just apply identity 1 (my last entry)... you'll arrive at...
cos^4x/sin^4x = cot^4x.... i'm sorry i have to go....

anonymous · Answer

Thanks I guess.  I know I have to end up a (cosx/sinx)^4

anonymous · Answer

we have a moderator viewing he can finish it....

anonymous · Answer

afk?

anonymous · Answer

ok i'm back... i miss my service....

anonymous · Answer

sorry that you missed it then.

anonymous · Answer

$$\frac{ \cos ^{2}x-\sin ^{2}x(1-\sin ^{2}x) }{ \sin ^{4}x }=\frac{ \cos ^{2}x-\sin ^{2}x \cos ^{2}x }{ \sin ^{4}x }$$

anonymous · Answer

factor out cos^2x....

anonymous · Answer

$$\frac{ \cos ^{2}x(1-\sin ^{2}x) }{ \sin ^{4}x }=\frac{ \cos ^{2}x \cos ^{2}x }{ \sin ^{4}x }=\frac{ \cos ^{4}x }{ \sin ^{4}x }$$

anonymous · Answer

$$\frac{ \cos ^{4}x }{ \sin ^{4}x }=\cot ^{4}x$$

anonymous · Answer

that completes the proof.... :-)

anonymous · Answer

I've been on this for 2 hours :<
 Thank you very much.

anonymous · Answer

you're welcome... anytime... :)