Question

find the exact value of the expression.
cos 75°

cos 75° =
(simplify your answer, including any radicals. use intege

Explanation:

Step1: Use the cosine addition formula

The formula is $\cos(A + B)=\cos A\cos B-\sin A\sin B$. We can write $75^{\circ}=45^{\circ}+30^{\circ}$. So, $\cos75^{\circ}=\cos(45^{\circ}+30^{\circ})$.

Step2: Substitute values of trigonometric functions

We know that $\cos45^{\circ}=\frac{\sqrt{2}}{2}$, $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, $\sin45^{\circ}=\frac{\sqrt{2}}{2}$, and $\sin30^{\circ}=\frac{1}{2}$.
Substituting these values into the formula:

$$
LATEXBLOCK0
$$

Answer:

$\frac{\sqrt{6}-\sqrt{2}}{4}$