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