Expert: Paul Klarreich - 3/10/2007

Question
Hi
I've been trying to work these problems...how do i prove these identities?

cos^4 x+1-sin^4 x=2cos^2 x

and

cscx/1+cscx-cscx/1-cscx=2sec^2 x

Answer
Questioner:   Britney
Category:  Advanced Math
 
Subject:  identities
Question:  Hi
I've been trying to work these problems...how do i prove these identities?

cos^4 x+1-sin^4 x=2cos^2 x

and

cscx/1+cscx-cscx/1-cscx=2sec^2 x
..............................................
Hi, Britney,

For your first one:

cos^4(x) + 1 - sin^4(x) = 2 cos^2(x)

cos^4(x) - sin^4(x) + 1 = 2 cos^2(x)

[Use C = cos x,  S = sin x, to save typing and preserve sanity.]

(C^2 + S^2)(C^2 - S^2) + 1 = 2 C^2

(    1    ) (C^2 - S^2) + 1 = 2 C^2

           C^2 + 1 - S^2   = 2 C^2

           C^2 + C^2     = 2 C^2

               2 C^2     = 2 C^2

Voila! Or Q.E.D., or whatever.
..............................
I assume the second one says:

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

Use csc x = 1/S on the left.
 1             1
---           ---
 S             S
-------   -  -------   = 2 sec^2(x)
    1            1
1 + ---      1 - ---  
    S            S

Clear fractions on the left; multiply each term by S.

 1         1
------ - ------ = 2 sec^2(x)
S + 1     S - 1

The opposite of S-1 is 1-S, so that will kill the (-) sign.

 1         1
------- + ------ = 2 sec^2(x)
1 + S     1 - S

Combine fractions on the left. LCD = (1 + S)(1 - S)


1 - S + 1 + S
--------------- = 2 sec^2(x)
(1 + S)(1 - S)

1 + 1
--------- = 2 sec^2(x)
1 - S^2

  2
--------- = 2 sec^2(x)
cos^2(x)

2 sec^2(x) = 2 sec^2(x)

Likewise.