Question

How do you prove `(sinx + cosx)^2=1 + sin2x`?

Answer 1

It depends on how far back you need to go in your proof.
If you can use:
`sin^2(x)+cos^2(x)=1` (which can be derived from the Pythagorean Theorem)
and `sin(2x) =2sin(x)*cos(x)` (the Double Angle Formula for sin, which is considerably more complicate to prove)
then the requested proof is fairly simple:

`(sin(x)+cos(x))^2`

`= sin^2(x) + 2sin(x)*cos(x) +cos^2(x)`

`= [sin^2(x)+cos^2(x)] + 2 sin(x)*cos(x)`

`= 1 + sin(2x)`