SOLUTION: Prove the following identity 2/1+cosx - tan^2 x/2 = 1

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Question 821540: Prove the following identity
2/1+cosx - tan^2 x/2 = 1
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
2%2F%281%2Bcosx%29+-+tan%5E2%28x%2F2%29+=+1
There are three variations of tan((1/2)x):
  • tan%28%281%2F2%29x%29 = +sqrt%28%281-cos%28x%29%29%2F%281%2Bcos%28x%29%29%29
  • tan%28%281%2F2%29x%29+=+sin%28x%29%2F%281%2Bcos%28x%29%29
  • tan%28%281%2F2%29x%29+=+%281-cos%28x%29%29%2Fsin%28x%29
Since our tan is squared, I'm going to use the first one (and, since we're squaring it, the + is not needed):
2%2F%281%2Bcosx%29+-+%28sqrt%28%281-cos%28x%29%29%2F%281%2Bcos%28x%29%29%29%29%5E2+=+1
Simplifying:
2%2F%281%2Bcosx%29+-+%281-cos%28x%29%29%2F%281%2Bcos%28x%29%29+=+1
Changing the subtraction between the fractions into an equivalent addition (since this situation is a source of much confusion and many errors):
2%2F%281%2Bcosx%29+%2B++%28-+%281-cos%28x%29%29%29%2F%281%2Bcos%28x%29%29+=+1
which simplifies to:
2%2F%281%2Bcosx%29+%2B++%28-1%2Bcos%28x%29%29%2F%281%2Bcos%28x%29%29+=+1
The denominators are equal so we can add the fractions:
%281%2Bcos%28x%29%29%2F%281%2Bcos%28x%29%29+=+1
And since the numerator and denominator are the same:
1=1