Question

Evaluate the following expression in the simplest form:
$\cos {{60}^{0}}\cos {{30}^{0}}-\sin {{60}^{0}}\sin {{30}^{0}}$

Answer 1

Hint: To solve the given expression in the simplest form we will use the trigonometric formulas that are needed here for that we have to analyze that what formula we can use for the given expression in the question or is it already in the form of some formula as that is the only way to solve it quickly. After using the formula we will arrange some terms to get our final answer.

Complete step-by-step solution
So, let’s begin our solution
If we look closely the given expression in the question is already in the form of a formula.
The formula is:
$\cos \left( x+y \right)=\cos x\cos y-\sin x\sin y$
So, now we have seen that the two expressions are similar,
We just have to put the values of x and y as per the question and from that, we can solve this expression in the simplest way.
As per the question,
$\begin{align}
  & x={{60}^{0}} \\
 & y={{30}^{0}} \\
\end{align}$
Now the given expression has been reduced to: $\cos {{90}^{0}}$
And the value of $\cos {{90}^{0}}$ is 0.
Hence the value of expression $\cos {{60}^{0}}\cos {{30}^{0}}-\sin {{60}^{0}}\sin {{30}^{0}}$ given in the question is 0.

Note: We can also solve this question by putting the values of $\sin {{30}^{0}},\cos {{30}^{0}},\sin {{60}^{0}},\cos {{60}^{0}}$ in the given expression and then we have to subtract those two get the final answer. As it will take time, but in the above method that much effort is not needed.