Question

find the exact value of the expression given below
cos (105°)

cos (105°)=□
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denomn

Explanation:

Step1: Use the cosine addition formula

The formula is $\cos(A + B)=\cos A\cos B-\sin A\sin B$. We can write $105^{\circ}=60^{\circ}+45^{\circ}$. So, $\cos(105^{\circ})=\cos(60^{\circ}+45^{\circ})$.

Step2: Substitute the values of trigonometric functions

We know that $\cos60^{\circ}=\frac{1}{2}$, $\cos45^{\circ}=\frac{\sqrt{2}}{2}$, $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, and $\sin45^{\circ}=\frac{\sqrt{2}}{2}$.
Substituting these values into the formula:

$$
LATEXBLOCK0
$$

Answer:

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