Prove that 1 + tan^2 x = sec^2 x
We know that tan x = sin x/cos x and so tan2x = sin2x/cos2x. We also know that sin2x + cos2x = 1 because this is a Pythagorean identity. We can rewrite the left hand side as (cos2x + sin2x)/cos2x because 1 can be rewritten as cos2x/cos2x. Because sin2x + cos2x = 1, we can simplify the numerator of the left hand side, meaning that (cos2x + sin2x)/cos2x = 1/cos2x which is sec2x (the right hand side). Therefore LHS=RHS and we have proven 1 + tan2 x = sec2 x
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