Prove that (sinx + cosx)^2 = 1 + 2sinxcosx

Starting on the left hand side we can expand out the brackets to get:
(sinx + cosx)(sinx + cosx)
sin²x+sinxcosx+sinxcosx+cos²x
Grouping together the like terms we can rearrange it to be:
sin²x + cos²x + 2sinxcosx
We now have one of the terms on the right hand side. We only need to get the 1. If we remember our identity sin²x + cos²x = 1 we can remove the first two terms and replace them with a 1, giving us:
1 + 2sinxcosx, the same as the right hand side, therefore proving the two are equal