Simplify (1+cos(2x))^2

${(1 + \cos (2 x))}^{2}$

Step 1

Rewrite ${(1 + \cos (2 x))}^{2}$ as $(1 + \cos (2 x)) (1 + \cos (2 x))$ .

$(1 + \cos (2 x)) (1 + \cos (2 x))$

Step 2

Expand $(1 + \cos (2 x)) (1 + \cos (2 x))$ using the FOIL Method.

Apply the distributive property.

$1 (1 + \cos (2 x)) + \cos (2 x) (1 + \cos (2 x))$

Apply the distributive property.

$1 \cdot 1 + 1 \cos (2 x) + \cos (2 x) (1 + \cos (2 x))$

Apply the distributive property.

$1 \cdot 1 + 1 \cos (2 x) + \cos (2 x) \cdot 1 + \cos (2 x) \cos (2 x)$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$1 + 1 \cos (2 x) + \cos (2 x) \cdot 1 + \cos (2 x) \cos (2 x)$

Multiply $\cos (2 x)$ by $1$ .

$1 + \cos (2 x) + \cos (2 x) \cdot 1 + \cos (2 x) \cos (2 x)$

Multiply $\cos (2 x)$ by $1$ .

$1 + \cos (2 x) + \cos (2 x) + \cos (2 x) \cos (2 x)$

Multiply $\cos (2 x) \cos (2 x)$ .

Raise $\cos (2 x)$ to the power of $1$ .

$1 + \cos (2 x) + \cos (2 x) + \cos^{1} (2 x) \cos (2 x)$

Raise $\cos (2 x)$ to the power of $1$ .

$1 + \cos (2 x) + \cos (2 x) + \cos^{1} (2 x) \cos^{1} (2 x)$

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

$1 + \cos (2 x) + \cos (2 x) + {\cos (2 x)}^{1 + 1}$

Add $1$ and $1$ .

$1 + \cos (2 x) + \cos (2 x) + \cos^{2} (2 x)$

Add $\cos (2 x)$ and $\cos (2 x)$ .

$1 + 2 \cos (2 x) + \cos^{2} (2 x)$