SOLUTION: Verify the identity. 1/1-cos + 1/1+cos = 2 Csc^2

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Question 814827: Verify the identity. 1/1-cos + 1/1+cos = 2 Csc^2
Found 2 solutions by mananth, DrBeeee:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
%281%2F%281-cosx%29%29+%2B+%281%2F%281%2Bcosx%29%29+=+2+Csc%5E2x+

LHS
%281-cosx%2B1-cosx%29%2F%281-Cos%5E2x%29

2/sin^2x

2csc%5E2x

Answer by DrBeeee(684) About Me  (Show Source):
You can put this solution on YOUR website!
Given;
(1) 1%2F%281-cosx%29+%2B+1%2F%281%2Bcosx%29+=+2csc%5E2%28x%29
First combine the fractions on the left side of (1) by cross multiplying to get
(2)
Now add the numerator terms and multiply the denominator factors of the left side of (2) to get
(3) %281%2B1%2Bcosx-cosx%29%2F%281%2Bcosx-cosx-cos%5E2%28x%29%29+=+2csc%5E2%28x%29 or
(4)%282%29%2F%281-cos%5E2%28x%29%29+=+2csc%5E2%28x%29
Now use the identity
(5) sin%5E2%28x%29+%2B+cos%5E2%28x%29+=+1 to get
(6) sin%5E2%28x%29+=+1+-+cos%5E2%28x%29
Now substitute (6) into (4) to get
(7)%282%29%2F%28sin%5E2%28x%29%29+=+2csc%5E2%28x%29
Now use the reciprocal identity
(8)1%2F%28sin%28x%29%29+=+csc%28x%29 to get
(9)1%2F%28sin%5E2%28x%29%29+=+csc%5E2%28x%29 or
by multiplying both sides of (9) by 2 we get
(10) 2%2F%28sin%5E2%28x%29%29+=+2csc%5E2%28x%29
Now substitute (10) into (7) to get
(11) 2csc%5E2%28x%29+=+2csc%5E2%28x%29 Q.E.D.