Question

How do you use an addition or subtraction formula to write the expression as a trigonometric function of one number and then find the exact value given $\cos \left( 80 \right)\cos \left( 10 \right)-\sin \left( 80 \right)\sin \left( 10 \right)$ ?

Answer 1

Hint: For these kinds of questions, we are supposed to make use of trigonometric formulae and equations. Let us see the expansion of $\cos \left( A+B \right)$. It is $\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B$. This is how we subtract or add trigonometric functions and shrink them. This is the addition method. Let us look at the subtraction i.e $\cos \left( A-B \right)$. It is $\cos \left( A-B \right)=\cos A\cos B+\sin A\sin B$. Let us use these concepts to find out the value of the given question.

Complete step by step solution:
In the question, we are given $\cos \left( 80 \right)\cos \left( 10 \right)-\sin \left( 80 \right)\sin \left( 10 \right)$. We certainly can not use the subtraction formula of $\cos $ since we can clearly see a positive symbol (+) in the given formula.
The formula is as follows :
$\Rightarrow \cos \left( A-B \right)=\cos A\cos B+\sin A\sin B$
 So we have to use the addition formula as there is a negative symbol that we can clearly see in the given formula and in the given question.
The formula is as follows :
$\Rightarrow \cos \left( A+B \right)=\cos A\cos B-\sin A\sin B$ .
Let us compare the given question with the formula that we wrote so that we can understand our $A$ and $B$.
Upon comparing with the formula, we get the following :
$\begin{align}
  & \Rightarrow A=80 \\
 & \Rightarrow B=10 \\
\end{align}$
Let us shrink the given question using the formula.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow \cos \left( 80 \right)\cos \left( 10 \right)-\sin \left( 80 \right)\sin \left( 10 \right) \\
 & \Rightarrow \cos \left( 80 \right)\cos \left( 10 \right)-\sin \left( 80 \right)\sin \left( 10 \right)=\cos \left( 80+10 \right)=\cos \left( 90 \right) \\
\end{align}$
We all know that $\cos \left( 90 \right)=0$.
Let us substitute that and find the answer.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow \cos \left( 80 \right)\cos \left( 10 \right)-\sin \left( 80 \right)\sin \left( 10 \right) \\
 & \Rightarrow \cos \left( 80 \right)\cos \left( 10 \right)-\sin \left( 80 \right)\sin \left( 10 \right)=\cos \left( 80+10 \right)=\cos \left( 90 \right)=0 \\
\end{align}$

Note: We should remember all the trigonometric formulae and equations as they all come in handy. We can also do this question by converting each product into sum by using $\cos C+\cos D=2\cos \left( \dfrac{C+D}{2} \right)\cos \left( \dfrac{C-D}{2} \right)$ and also $\sin C+\sin D=2\sin \left( \dfrac{C+D}{2} \right)\cos \left( \dfrac{C-D}{2} \right)$ .
This method can be a little lengthy and confusing. We should be very careful while substituting the value. Here all of our angles are in degrees