find the exact value of each expression. (a) cos75° + cos15° = (b) sin105°cos15° =

Explanation:

Step1: Use sum - to - product formula for (a)

The sum - to - product formula is $\cos A+\cos B = 2\cos\frac{A + B}{2}\cos\frac{A - B}{2}$. For $\cos75^{\circ}+\cos15^{\circ}$, where $A = 75^{\circ}$ and $B=15^{\circ}$, we have $2\cos\frac{75^{\circ}+15^{\circ}}{2}\cos\frac{75^{\circ}-15^{\circ}}{2}=2\cos45^{\circ}\cos30^{\circ}$. Since $\cos45^{\circ}=\frac{\sqrt{2}}{2}$ and $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, then $2\times\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}=\frac{\sqrt{6}}{2}$.

Step2: Rewrite angles for (b)

We know that $\sin105^{\circ}=\sin(90^{\circ}+15^{\circ})=\cos15^{\circ}$. So $\sin105^{\circ}\cos15^{\circ}=\cos15^{\circ}\cos15^{\circ}=\cos^{2}15^{\circ}$. Using the half - angle formula $\cos^{2}\alpha=\frac{1 + \cos2\alpha}{2}$, with $\alpha = 15^{\circ}$ and $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, we get $\frac{1+\cos30^{\circ}}{2}=\frac{1+\frac{\sqrt{3}}{2}}{2}=\frac{2 + \sqrt{3}}{4}$.

Answer:

(a) $\frac{\sqrt{6}}{2}$ (b) $\frac{2+\sqrt{3}}{4}$