Question

Use sum, difference, product, or half-angle formulas to find the exact value of the expression cos15$°$

Short answer

$\cos 15°=\frac{\sqrt{3}+1}{2\sqrt{2}}$

Step-by-step solution

Step 1. Rewriting the expression

We know that ,

cos(A-B) = cos(A)cos(B) +sin(A)sin(B)

15$°$can be written as 45$°$-30$°$

So,

$$\begin{gathered} \cos 15°=\cos (45°-30°) \\ \cos 15°=\cos 45°\cos 30°+\sin 45°\sin 30° \end{gathered}$$

Step 2. Calculations

$$\begin{gathered} \cos 15°=\cos 45°\cos 30°+\sin 45°\sin 30° \\ \cos 15°=\frac{1}{\sqrt{2}}\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\frac{1}{2} \\ \cos 15°=\frac{\sqrt{3}+1}{2\sqrt{2}} \end{gathered}$$