# How do you prove 1 + sin 2x = (sin x + cos x)^2?

Feb 19, 2016

#### Explanation:

Remember : ${\sin}^{2} x + {\cos}^{2} x = 1$

$2 \sin x \cos x = \sin 2 x$

Step 1: Rewrite the problem as it is

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

Step 2: Pick a side you want to work on - (right hand side is more complicated)

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

$= {\sin}^{2} x + \sin x \cos x + \sin x \cos x + {\cos}^{2} x$

$= {\sin}^{2} x + 2 \sin x \cos x + {\cos}^{2} x$

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

$= 1 + 2 \sin x \cos x$

= $1 + \sin 2 x$

Q.E.D

Noted: the left hand side is equal to right hand side, this meant this expression is correct. We can conclude the proof by add QED (in Latin meant quod erat demonstrandum, or "which is what had to be proven")