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

Mar 8, 2016

This result follows almost directly from the following:

• ${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$
• ${\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$
• $\sin \left(2 x\right) = 2 \sin \left(x\right) \cos \left(x\right)$

With these, we have

${\left(\sin \left(x\right) + \cos \left(x\right)\right)}^{2} = {\sin}^{2} \left(x\right) + 2 \sin \left(x\right) \cos \left(x\right) + {\cos}^{2} \left(x\right)$

$= \left({\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right)\right) + 2 \sin \left(x\right) \cos \left(x\right)$

$= 1 + \sin \left(2 x\right)$