SOLUTION: prove the following identity: (1-tanA)^2=sec^2A-2tanA

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Question 618397: prove the following identity: (1-tanA)^2=sec^2A-2tanA
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%281-tan%28A%29%29%5E2=sec%5E2%28A%29-2tan%28A%29
Since the right side is not in factored form like the left side, let's multiply out the left side. FOIL can be used to do this but I prefer using the %28a-b%29%5E2+=+a%5E2-2ab%2Bb%5E2 pattern:
%281%29%5E2-2%281%29%28tan%28A%29%29%2B%28tan%28A%29%29%5E2=sec%5E2%28A%29-2tan%28A%29
which simplifies to:
1-2tan%28A%29%2Btan%5E2%28A%29=sec%5E2%28A%29-2tan%28A%29

At this point we should see that we have -2tan(A) on both sides of the equation. What is different is that the left side has 1+%2B+tan%5E2%28A%29 and the right side has sec%5E2%28A%29. SO if we could find a way to change 1+%2B+tan%5E2%28A%29 into sec%5E2%28A%29 then we'd be done.

If you know your properties well you know that 1+%2B+tan%5E2%28A%29 is always equal to sec%5E2%28A%29. So we can just substitute one for the other:
sec%5E2%28A%29-2tan%28A%29+=+sec%5E2%28A%29-2tan%28A%29

P.S. There appears to be a multiplication symbol between the "sec^2" and the "(A)". It should not be there. This is a fault in algebra.com's formula drawing software.