SOLUTION: Prove that [ 1 + sin(2x)] / [ 1 + cos(2x)] = 1/2 (1+ tan(x))^2

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Question 1142320: Prove that
[ 1 + sin(2x)] / [ 1 + cos(2x)] = 1/2 (1+ tan(x))^2
Answer by math_helper(2461) About Me  (Show Source):
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LHS:
+%28+1+%2B+sin%282x%29%29+%2F+%28+1+%2B+cos%282x%29%29+

Use +1%2Bsin%282x%29+=+1%2B2sin%28x%29cos%28x%29+ and +cos%282x%29+=+cos%5E2%28x%29-sin%5E2%28x%29 :

= +%28+1+%2B+2%2Asin%28x%29cos%28x%29%29+%2F+%28+1+%2B+cos%5E2%28x%29+-+sin%5E2%28x%29%29+

= +%28+1+%2B+2%2Asin%28x%29cos%28x%29%29+%2F+%28+2%2Acos%5E2%28x%29%29+

= +%281%2F2%29+%28%281%2Fcos%5E2%28x%29%29+%2B+2%2A%28sin%28x%29%2Fcos%28x%29%29%29+

= +%281%2F2%29+%28+sec%5E2%28x%29+%2B+2tan%28x%29%29+


RHS:
+%281%2F2%29+%281%2B+tan%28x%29%29%5E2++

Expand:
= ++%281%2F2%29+%281+%2B+2tan%28x%29+%2B+tan%5E2%28x%29%29+

Use: +1%2Btan%5E2%28x%29+=+sec%5E2%28x%29+:

= +%281%2F2%29+%28sec%5E2%28x%29+%2B+2tan%28x%29%29+

The LHS and RHS are both equal to +%281%2F2%29+%28sec%5E2%28x%29+%2B+2tan%28x%29%29+ and hence they are both equal to each other.