Question

How do you verify `(sin x+cos x)^2=1+sin2x`?

Answer 1

This result follows almost directly from the following:

• `(a+b)^2 = a^2+2ab + b^2`
• `sin^2(x) + cos^2(x) = 1`
• `sin(2x) = 2sin(x)cos(x)`

With these, we have

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

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

`=1 + sin(2x)`