## anonymous one year ago (_____)^2 = (csc x-1)(csc x+1) What is suppose to be in the blank? ***I know this is a trig identity and do not have my sheet with me. No online refrences are helping me***

1. anonymous

I know it has to be either tan x or cot x. My memory is not on my side today.

2. UsukiDoll

ok.. there are three trig identities. the most common is $\cos^2x+\sin^2x=1$

3. UsukiDoll

but there are two more equations.

4. anonymous

Ok

5. UsukiDoll

$1+\tan^2x=\sec^2x$ $1+\cot^2x = \csc^2x$

6. UsukiDoll

let's solve the right hand side of the equation to see who is the culprit lol xD so we expand $(cscx-1)(cscx+1)$

7. anonymous

Haha ok

8. UsukiDoll

just use foil ... you will notice that O and I cancel out

9. anonymous

Not familiar with FOIL (Sorry). My teacher calls it the "F word" (lol)

10. UsukiDoll

first outer inner last

11. UsukiDoll

$(cscx-1)(cscx+1) = (cscx)(cscx)+(1)(cscx)+(-1)(cscx)+(1)(1)$

12. UsukiDoll

fffffffff missed a sign on the last one should be +(-1)(1)

13. anonymous

Oh ok! I think I see what you are getting at

14. UsukiDoll

$(cscx-1)(cscx+1) = (cscx)(cscx)+(1)(cscx)+(-1)(cscx)+(-1)(1) = (cscx)^2+cscx-cscx-1$

15. UsukiDoll

OY ! $(cscx)^2+cscx-cscx-1$

16. UsukiDoll

the cscx-cscx cancels out

17. UsukiDoll

$\csc^2x-1$

18. anonymous

Then we can add the 1 right?

19. UsukiDoll

do you remember $1+\cot^2x = \csc^2x ?$

20. UsukiDoll

what do I need to do have $\cot^2x$ by itself

21. anonymous

Yep!

22. anonymous

subract the 1?

23. UsukiDoll

yes subtract on both sides..

24. anonymous

Oh ok! So we just proved that cot^2 (x) is my answer right?

25. UsukiDoll

yeah

26. UsukiDoll

$\cot^2x = \csc^2x-1$

27. anonymous

Would I just write it as cotx?

28. UsukiDoll

huh? you mean for (cotx)^2 = (csc^2x-1)

29. anonymous

Yes

30. UsukiDoll

hmmm... if you don't forget this part $\cot^2x = \csc^2x-1$

31. UsukiDoll

$(cotx)^2 = (cscx)^2-1$ means the same

32. UsukiDoll

just don't write cot2x and csc2x-1 BIG NO NO!

33. anonymous

Ok! Thank you for your time and teaching me!

34. anonymous

if u observe the RHS, u see it results csc^2-1 there is also an identity saying that, cosec^2-cot^2=1 that implies cosec^2-1=cot^2...therefore theRHS is cot^2 therefore obviously LHS must be cot^2...therefore the answer is cot

35. UsukiDoll

36. UsukiDoll

the term cosecant isn't used for the Pythagorean identities.

37. UsukiDoll

but there is one identity that is used beyond trig and that's the $\cos^2x+\sin^2x = 1$