Question

Find $\cos 60\times \cos 30+\sin 60\times \sin 30$ .

Answer 1

Hint : We first describe the concept and formulas of compound angles for the trigonometric ratio cos. We use the formulas of $\cos A\cos B+\sin A\sin B=\cos \left( A-B \right)$ . We put the values of $A=60;B=30$ to find the solution for $\cos 60\times \cos 30+\sin 60\times \sin 30$ .

Complete step-by-step answer :
To simplify or to express in product form the given expression
$\cos 60\times \cos 30+\sin 60\times \sin 30$ , we are going to use the laws of compound angles. A compound angle is an algebraic sum of two or more angles. We use trigonometric identities to connote compound angles through trigonometric functions.
The formula for compound angles gives
$\cos A\cos B+\sin A\sin B=\cos \left( A-B \right)$ .
We assume the variables as $A=60;B=30$ .
Putting the values, we get
$\cos 60\cos 30+\sin 60\sin 30=\cos \left( 60-30 \right)$ .
The simplified form is $\cos 60\cos 30+\sin 60\sin 30=\cos 30$
We know that the value for $\cos 30=\dfrac{\sqrt{3}}{2}$ .
Therefore, we have $\cos 60\times \cos 30+\sin 60\times \sin 30=\dfrac{\sqrt{3}}{2}$ .
So, the correct answer is “ $\dfrac{\sqrt{3}}{2}$ ”.

Note : We can also put the individual values of the terms in $\cos 60\times \cos 30+\sin 60\times \sin 30$ .
We know that $\cos 60=\sin 30=\dfrac{1}{2},\cos 30=\sin 60=\dfrac{\sqrt{3}}{2}$ .
Putting the values, we get $\cos 60\times \cos 30+\sin 60\times \sin 30=\dfrac{1}{2}\times \dfrac{\sqrt{3}}{2}+\dfrac{\sqrt{3}}{2}\times \dfrac{1}{2}=\dfrac{\sqrt{3}}{2}$ .