Simplify (rcos(theta)-2)^2

${(r \cos (θ) - 2)}^{2}$

Rewrite ${(r \cos (θ) - 2)}^{2}$ as $(r \cos (θ) - 2) (r \cos (θ) - 2)$ .

$(r \cos (θ) - 2) (r \cos (θ) - 2)$

Expand $(r \cos (θ) - 2) (r \cos (θ) - 2)$ using the FOIL Method.

Apply the distributive property.

$r \cos (θ) (r \cos (θ) - 2) - 2 (r \cos (θ) - 2)$

Apply the distributive property.

$r \cos (θ) (r \cos (θ)) + r \cos (θ) \cdot - 2 - 2 (r \cos (θ) - 2)$

Apply the distributive property.

$r \cos (θ) (r \cos (θ)) + r \cos (θ) \cdot - 2 - 2 (r \cos (θ)) - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $r$ by $r$ by adding the exponents.

Move $r$ .

$r \cdot r \cos (θ) \cos (θ) + r \cos (θ) \cdot - 2 - 2 (r \cos (θ)) - 2 \cdot - 2$

Multiply $r$ by $r$ .

$r^{2} \cos (θ) \cos (θ) + r \cos (θ) \cdot - 2 - 2 (r \cos (θ)) - 2 \cdot - 2$

Multiply $r^{2} \cos (θ) \cos (θ)$ .

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

$r^{2} (\cos^{1} (θ) \cos (θ)) + r \cos (θ) \cdot - 2 - 2 (r \cos (θ)) - 2 \cdot - 2$

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

$r^{2} (\cos^{1} (θ) \cos^{1} (θ)) + r \cos (θ) \cdot - 2 - 2 (r \cos (θ)) - 2 \cdot - 2$

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

$r^{2} {\cos (θ)}^{1 + 1} + r \cos (θ) \cdot - 2 - 2 (r \cos (θ)) - 2 \cdot - 2$

Add $1$ and $1$ .

$r^{2} \cos^{2} (θ) + r \cos (θ) \cdot - 2 - 2 (r \cos (θ)) - 2 \cdot - 2$

Move $- 2$ to the left of $r \cos (θ)$ .

$r^{2} \cos^{2} (θ) - 2 \cdot (r \cos (θ)) - 2 (r \cos (θ)) - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$r^{2} \cos^{2} (θ) - 2 r \cos (θ) - 2 r \cos (θ) + 4$

Subtract $2 r \cos (θ)$ from $- 2 r \cos (θ)$ .

$r^{2} \cos^{2} (θ) - 4 r \cos (θ) + 4$