Trig Identity: Stumped for two hours! Please help!?

Question

I've been stumped on trying to prove the following trig identity for the last two hours, I got the other nine on my worksheet after considerable trial and error. :) Can someone please give me a hand with this monster? Thanks:

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

Actually, I did find that the top equals -1. I'm stumped with the denominator on the LHS. I am unsure what to do. I've tried factoring out Cot, I've tried changing cot(csc) to cos/sin^2, I keep coming up with dead ends and getting flustered! :(

Answer 1

Left side:

{[cos^2(x)/sin^2(x) - 1/sin^2(x)]}/{cos^(x)/sin^2(x) - cos(x)/sin^2(x)}

sin^2(x) is denominator of all 4 terms, so combine on each side

= (cos^2(x)-1)/(cos^2(x) - cos(x)

= (cos^2(x)-1)/cos(x)(cos(x)-1)
Factor top as diff. of two squares;

= [(cos(x) + 1) (cos(x)-1)]/cos(x)(cos(x)-1)

Cancel factor of (cos(x)-1)

= (cos(x)+1)/cos(x)

= cos(x)/cos(x) + 1/cos(x) = 1 + sec(x)

Answer 2

i just did it, it really does work. try changing everthing to sin and cos, sometimes that helps me. to get you started cot^2 - csc^2 = -1.

Answer 3

I got it!
You start out with cot^2(x) - csc^2(x)) =-1
Then you have -1/(cot^2(x) - cot(x)csc(x))

(cot^2(x) - cot(x)csc(x)) =cos squared/sin squared- (cos/sin)(1/sin)

=-1/ ((cos squared-cos)/sin squared)

You can figure out the rest!
Good luck!

Answer 4

My brain hurts. Sorry, now I have to lie down.