Question

Find the value of $x$ if $\cos x = \cos {60^\circ}\cos {30^\circ} + \sin {60^\circ}\sin {30^\circ}$

Answer 1

According to given in the question we have to find the value of $x$ when $\cos x = \cos {60^\circ}\cos {30^\circ} + \sin {60^\circ}\sin {30^\circ}$ so, first of all we have to know about the formula of $\cos A\cos B + \operatorname{Sin} A\sin B$ that is mentioned below:

Formula used: $\cos A\cos B + \operatorname{Sin} A\sin B$= $\cos (A - B)....................(A)$
Now with the help of the formula (A) as mentioned above, we have to substitute all the values in the formula (A) and then we have to
The, we compare $\cos x = \cos (A - B)$ to get the desired value of $x$

Complete step-by-step solution:
Step 1: As mentioned in the question that $\cos {60^\circ}\cos {30^\circ} + \sin {60^\circ}\sin {30^\circ}$ is in the form of $\cos A\cos B + \operatorname{Sin} A\sin B$ where $A = {60^\circ}$ and $B = {30^\circ}$
Step 2: Hence, we can apply the formula $(A)$ in which is as mentioned in the solution step 1,
$\Rightarrow \cos x = \cos ({60^\circ} - {30^\circ}) \\\ \Rightarrow \cos x = \cos {30^\circ}$
Step 3: Now, comparing the L.H.S and R.H.S obtained in the solution step 2.
Hence, we get,
$\Rightarrow x = {30^\circ}$

Hence, the value of $x = {30^\circ}$ for the given expression $\cos x = \cos {60^\circ}\cos {30^\circ} + \sin {60^\circ}\sin {30^\circ}$

Note: To determine the value of the given trigonometric expression it is necessary that we have to know about the formula of $\cos A\cos B + \operatorname{Sin} A\sin B = \cos (A - B)$
To determine the value of x it is necessary that we have to compare it with the obtained trigonometric expression after applying the formula.