Chapter 7, Problem 77RE

To determine

To find:the exact value of the given expression.

Answer to Problem 77RE

  $\cos {80}^{\circ }\cos {20}^{\circ }+\sin {80}^{\circ }\sin {20}^{\circ }=\frac{1}{2}$ .

Explanation of Solution

Given:

The given expression is $\cos {80}^{\circ }\cos {20}^{\circ }+\sin {80}^{\circ }\sin {20}^{\circ }$ .

Calculation:

As the given expressionis $\cos {80}^{\circ }\cos {20}^{\circ }+\sin {80}^{\circ }\sin {20}^{\circ }$ .

Now, using the identity $\cos A\cos B+sinA\sin B=\cos \left(A-B\right)$ :

  $\begin{array}{l}\cos {80}^{\circ }\cos {20}^{\circ }+\sin {80}^{\circ }\sin {20}^{\circ }=\cos \left(80-20\right)\\ =\cos \left({60}^{\circ }\right)=\frac{1}{2}\end{array}$

Hence, the value of $\cos {80}^{\circ }\cos {20}^{\circ }+\sin {80}^{\circ }\sin {20}^{\circ }=\frac{1}{2}$ .