Expert: Sherman D. - 4/14/2009

Question
I need help proving 2 identities.

1) csc^2 A/2 = 2sec A/sec A - 1

2) 4sin^4 A/2 = 1 - 2cosA + cos^2 A

thanks for the help.

Answer
if by this you mean

1.)
csc(A/2)^2 = (2sec(A))/(sec(A) - 1)

same as saying

1/(sin(A/2)^2) = (2(1/cos(A)))/((1/cos(A)) - 1)

sin(A/2)^2 = (1 - cos(A))/2

1/((1 - cos(A))/2) = (2/cos(A))/((1 - cos(A))/cos(A))

2/(1 - cos(A)) = (2cos(A))/(cos(A)(1 - cos(A)))

the cos(A)s cancel out on the left side

2/(1 - cos(A)) = 2/(1 - cos(A))

so as you can see both sides are equal.

-------------------------------------------------------

2.)
4sin(A/2)^2 = 1 - 2cos(A) + cos(A)^2
4((1 - cos(A))/2) = 1 - 2cos(A) + cos(A)^2
2(1 - cos(A)) = 1 - 2cos(A) + cos(A)^2
2(-cos(A) + 1) = cos(A)^2 - 2cos(A) + 1
-2(cos(A) - 1) = (cos(A) - 1)^2

divide both sides by cos(A) - 1

-2 = cos(A) - 1
cos(A) = -1
A = 180 or pi

more info found at

http://depositfiles.com/files/go5uq3ws3

http://math2.org/math/trig/identities.htm