Question

How do you use sum to product formulas to write the sum or difference $\cos \left( x+\dfrac{\pi }{6} \right)-\cos \left( x-\dfrac{\pi }{6} \right)$ as a product?

Answer 1

Hint: We first describe the concept and formulas of compound angles for the trigonometric ratio cos. We use the formulas of $\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B$ and $\cos \left( A-B \right)=\cos A\cos B+\sin A\sin B$. We put the values of $A=x;B=\dfrac{\pi }{6}$ to find the solution for $\cos \left( x+\dfrac{\pi }{6} \right)-\cos \left( x-\dfrac{\pi }{6} \right)$.

Complete step-by-step answer:
To simplify or to express in product form the given expression $\cos \left( x+\dfrac{\pi }{6} \right)-\cos \left( x-\dfrac{\pi }{6} \right)$, we are going to use the laws of compound angles.
The formula for compound angles gives $\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B$ and $\cos \left( A-B \right)=\cos A\cos B+\sin A\sin B$.
Now we subtract these two formulas to get
$\begin{align}
  & \cos \left( A+B \right)-\cos \left( A-B \right) \\
 & =\cos A\cos B-\sin A\sin B-\cos A\cos B-\sin A\sin B \\
 & =-2\sin A\sin B \\
\end{align}$
For our given expression, we use the formula $\cos \left( A+B \right)-\cos \left( A-B \right)=-2\sin A\sin B$.
We assume the variables as $A=x;B=\dfrac{\pi }{6}$.
Putting the values, we get $\cos \left( x+\dfrac{\pi }{6} \right)-\cos \left( x-\dfrac{\pi }{6} \right)=-2\sin x\sin \left( \dfrac{\pi }{6} \right)$.
We know that the value for $\sin \left( \dfrac{\pi }{6} \right)=\dfrac{1}{2}$.
Therefore, we have $-2\sin x\sin \left( \dfrac{\pi }{6} \right)=-2\sin x\left( \dfrac{1}{2} \right)=-\sin x$.
Therefore, changing from sum to product we get the simplified form of $\cos \left( x+\dfrac{\pi }{6} \right)-\cos \left( x-\dfrac{\pi }{6} \right)$ as $-\sin x$.

Note: A compound angle is an algebraic sum of two or more angles. We use trigonometric identities to connote compound angles through trigonometric functions. Similarly, we can change the form from product to sum using the formula of
$2\sin A\sin B=\cos \left( A-B \right)-\cos \left( A+B \right)$.