Question

Expand the following expression $\cos x\cos 2x\cos 3x$:

Answer 1

Hint: To expand the given expression we will write the formula for cos 2x and cos 3x, and then we will multiply the whole expression and then we will rearrange some terms to get the final answer.

Complete step-by-step solution -
Now we are going to write the formula for both the expression:
$\begin{align}
  & \cos 2x=2{{\cos }^{2}}x-1 \\
 & \cos 3x=4{{\cos }^{3}}x-3\cos x \\
\end{align}$
 Now as we have written all the required formula we are going to put it in our equation and let’s see what we get,
$\cos x\cos 2x\cos 3x$
Putting the values in it we get,
$\cos x\left( 2{{\cos }^{2}}x-1 \right)\left( 4{{\cos }^{3}}x-3\cos \right)$
Now multiplying them we get,
$\begin{align}
  & \left( 2{{\cos }^{3}}x-\cos x \right)\left( 4{{\cos }^{3}}x-3\cos \right) \\
 & =8{{\cos }^{6}}x-6{{\cos }^{4}}x-4{{\cos }^{4}}x+3{{\cos }^{2}}x \\
 & =8{{\cos }^{6}}x-10{{\cos }^{4}}x+3{{\cos }^{2}}x \\
\end{align}$
Now we have expanded and rearranged the given equation in the question.
Hence, this is the final answer to this question.
One can use some method to check whether the given answer is true or not, like by putting some value of x and check whether it is true or not.
Let’s check the solution by putting x = 0 ,
$\begin{align}
  & \cos 0\times \cos 0\times \cos 0=8({{\cos }^{6}}0)-10{{\cos }^{4}}0+3{{\cos }^{2}}0 \\
 & 1=8-10+3 \\
 & 1=1 \\
\end{align}$
We get both the values equal to 1.
Hence, our solution is correct.

Note: The students might confuse writing the formula as for sin and cos they are complementary to each other and one should be careful while multiplying the values which is very important to get the correct answer, so always check by putting some values just to be sure.