How do you prove (sinx+cosx)^2 = 1+2sinxcosx?

Jun 20, 2015

Use trigonometric identities and the FOIL method.

Explanation:

We are asked to prove that ${\left(\sin x + \cos x\right)}^{2} = 1 + 2 \sin \left(x\right) \cos \left(x\right)$.

1) Change ${\left(\sin x + \cos x\right)}^{2}$ to $\left(\sin x + \cos x\right) \left(\sin x + \cos x\right)$ (since the square of any expression is that expression multiplied by itself.)

2) Utilize the FOIL method for multiplying binomials, e.g. $\left(\sin x + \cos x\right) \left(\sin x + \cos x\right) = \left(\sin x\right) \left(\sin x\right) + \left(\sin x\right) \left(\cos x\right) + \left(\cos x\right) \left(\sin x\right) + \left(\cos x\right) \left(\cos x\right)$

3) Simplify and group like terms: $\left(\sin x\right) \left(\sin x\right) + \left(\sin x\right) \left(\cos x\right) + \left(\cos x\right) \left(\sin x\right) + \left(\cos x\right) \left(\cos x\right) = {\sin}^{2} x + {\cos}^{2} x + 2 \sin x \cos x$

4) Recall the trigonometric identity which states ${\sin}^{2} x + {\cos}^{2} x = 1$, and substitute into (3): ${\sin}^{2} x + {\cos}^{2} x + 2 \sin x \cos x = 1 + 2 \sin x \cos x$

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