Question

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

Answer 1

You will be squaring a quantity that contains both `sinx` and `cosx`, so you will need the identity `sin^2x+cos^2x = 1`. I would start by multiplying both of these out.

`(1+sinx+cosx)^2 = 1 + sinx + cosx + sinx + sin^2x + sinxcosx + cosx + sinxcosx + cos^2x`

`= 1 + 2sinx + 2cosx + sin^2x + cos^2x + 2sinxcosx`

while

`2(1+sinx)(1+cosx) = 2(1+cosx+sinx+sinxcosx) = 2+2cosx+2sinx+2sinxcosx`

Compare:
`1 + sin^2x + cos^2x + cancel(2sinx + 2cosx + 2sinxcosx) = 2+cancel(2cosx+2sinx+2sinxcosx)`

`1 + sin^2x + cos^2x = 2`

`1 + 1 = 2`

`2 = 2`

They're equal.