Question

Find the value of $\cos {15}^{\circ }$using $\cos (A-B)$ identity

Answer 1

Find the value of $\cos {15}^{\circ }$ using the identity to find cosine of difference of angles

$cos{15}^{\circ }$ can be written as

$cos{15}^{\circ }=cos{(45–30)}^{\circ }$…………………..(1)

By using the identity

$cos(A–B)=cosA\times cosB+sinA\times sinB$

$cos{(45–30)}^{\circ }=cos{45}^{\circ }\times cos{30}^{\circ }+sin{45}^{\circ }\times sin{30}^{\circ }$…………………..(2)

Substitute the value of $\sin {30}^{\circ },\sin {45}^{\circ },\cos {30}^{\circ }$and $\cos {45}^{\circ }$ degrees, then equation (2) will become

$cos{(45–30)}^{\circ }=\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\times \frac{1}{2}$

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

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

Hence the value of $\cos {15}^{\circ }$ is $\frac{\sqrt{3}+1}{2\sqrt{2}}$