Find the Exact Value cos(arctan(- square root of 3))

$\cos (\arctan (- \sqrt{3}))$

Apply the product rule to $- \sqrt{3}$ .

$\frac{1}{\sqrt{1 + {(- 1)}^{2} {\sqrt{3}}^{2}}}$

Raise $- 1$ to the power of $2$ .

$\frac{1}{\sqrt{1 + 1 {\sqrt{3}}^{2}}}$

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$\frac{1}{\sqrt{1 + {\sqrt{3}}^{2}}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{\sqrt{1 + {(3^{\frac{1}{2}})}^{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{\sqrt{1 + 3^{\frac{1}{2} \cdot 2}}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{1}{\sqrt{1 + 3^{\frac{2}{2}}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{\sqrt{1 + 3^{\frac{2}{2}}}}$

Rewrite the expression.

$\frac{1}{\sqrt{1 + 3^{1}}}$

Evaluate the exponent.

$\frac{1}{\sqrt{1 + 3}}$

Add $1$ and $3$ .

$\frac{1}{\sqrt{4}}$

Rewrite $4$ as $2^{2}$ .

$\frac{1}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{2}$

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$