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} $ .