I think I am missing that bone in the brain that allows you to prove trigonometric identities. I just got done with the first half of a final, and I totally botched one of the questions, which was “Prove tan (pi / 4 + x) = (1 + sin2x) / cos2x”. I got as far as changing the left side to (cosx + sinx) / (cosx - sinx), but then I got stuck. Wasted a full sheet of paper trying different ways to solve it.

What really gets me is I know we’ve done this problem (or something similar) a couple of times before, but I’ve been unable to figure it out each time. I know all of the standard identities and half-angle/double-angle/addition/subtraction equations but I can’t make them work here. It’s probably something completely obvious, too.

Update: Yeah. It was obvious.

• tan (pi / 4 + x) = (tan (pi / 4) + tan (x)) / (1 - tan (pi / 4) tan (x))
• tan (pi / 4) = 1, so (1 + tan (x)) / (1 - tan (x))
• Multiply by cos (x) / cos (x): (cos (x) + sin (x)) / (cos (x) - sin (x))
• Multiply by the (cos (x) + sin (x)) / (cos (x) + sin (x))): (cos^2 (x) + 2 sin (x) cos (x) + sin^2 (x)) / (cos^2 (x) - sin^2 (x))
• cos^2 (x) + sin^2 (x) = 1, 2 sin(x) cos (x) = sin (2x), cos^2 (x) - sin^2 (x) = cos (2x), so:
• Finally we have LHS (1 + sin (2x)) / cos (2x), same as RHS.

All I needed was a few days to ruminate over it. Bah.