Evaluate (sin(theta)+cos(theta))^2

${(\sin (θ) + \cos (θ))}^{2}$

Rewrite ${(\sin (θ) + \cos (θ))}^{2}$ as $(\sin (θ) + \cos (θ)) (\sin (θ) + \cos (θ))$ .

$(\sin (θ) + \cos (θ)) (\sin (θ) + \cos (θ))$

Expand $(\sin (θ) + \cos (θ)) (\sin (θ) + \cos (θ))$ using the FOIL Method.

Apply the distributive property.

$\sin (θ) (\sin (θ) + \cos (θ)) + \cos (θ) (\sin (θ) + \cos (θ))$

Apply the distributive property.

$\sin (θ) \sin (θ) + \sin (θ) \cos (θ) + \cos (θ) (\sin (θ) + \cos (θ))$

Apply the distributive property.

$\sin (θ) \sin (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos (θ) \cos (θ)$

Simplify and combine like terms.

Simplify each term.

Multiply $\sin (θ) \sin (θ)$ .

Raise $\sin (θ)$ to the power of $1$ .

$\sin^{1} (θ) \sin (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos (θ) \cos (θ)$

Raise $\sin (θ)$ to the power of $1$ .

$\sin^{1} (θ) \sin^{1} (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos (θ) \cos (θ)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\sin (θ)}^{1 + 1} + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos (θ) \cos (θ)$

Add $1$ and $1$ .

$\sin^{2} (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos (θ) \cos (θ)$

Multiply $\cos (θ) \cos (θ)$ .

Raise $\cos (θ)$ to the power of $1$ .

$\sin^{2} (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos^{1} (θ) \cos (θ)$

Raise $\cos (θ)$ to the power of $1$ .

$\sin^{2} (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos^{1} (θ) \cos^{1} (θ)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin^{2} (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + {\cos (θ)}^{1 + 1}$

Add $1$ and $1$ .

$\sin^{2} (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos^{2} (θ)$

Reorder the factors of $\sin (θ) \cos (θ)$ .

$\sin^{2} (θ) + \cos (θ) \sin (θ) + \cos (θ) \sin (θ) + \cos^{2} (θ)$

Add $\cos (θ) \sin (θ)$ and $\cos (θ) \sin (θ)$ .

$\sin^{2} (θ) + 2 \cos (θ) \sin (θ) + \cos^{2} (θ)$

Move $\cos^{2} (θ)$ .

$\sin^{2} (θ) + \cos^{2} (θ) + 2 \cos (θ) \sin (θ)$

Apply pythagorean identity.

$1 + 2 \cos (θ) \sin (θ)$

Simplify each term.

Reorder $2 \cos (θ)$ and $\sin (θ)$ .

$1 + \sin (θ) (2 \cos (θ))$

Reorder $\sin (θ)$ and $2$ .

$1 + 2 \cdot \sin (θ) \cos (θ)$

Apply the sine double-angle identity.

$1 + \sin (2 θ)$