Question

How do I rewrite \[3\cos 4x\] in terms of \[\cos x\]?

Answer 1

Hint: In this problem we have to rewrite \[3\cos 4x\] in terms of \[\cos x\]. Here we have to use the trigonometric identity called ‘cosine double angle formula’ \[\cos \left( 2\theta \right)=2{{\cos }^{2}}\theta -1\] to rewrite the given trigonometric expression in terms of \[\cos x\]. We can then simplify step by step and we will use the formula twice to find the answer in terms of \[\cos x\].

Complete step by step solution:
We know that the given trigonometric expression to be rewritten in terms of \[\cos x\] is
\[3\cos 4x\].
We can write the above expression as,
\[\Rightarrow 3\cos \left( 2\left( 2x \right) \right)\]
We can now use the trigonometric identity called ‘cosine double angle formula’ \[\cos \left( 2\theta \right)=2{{\cos }^{2}}\theta -1\]
We can see that \[\theta =2x\], we can apply this formula to the above step, we get
\[\Rightarrow 3\left( 2{{\cos }^{2}}\left( 2x \right)-1 \right)\]
We can now multiply the number 3 inside the brackets, we get
\[\begin{align}
  & \Rightarrow 6{{\cos }^{2}}\left( 2x \right)-3 \\
 & \Rightarrow 6{{\left( \cos 2x \right)}^{2}}-3 \\
\end{align}\]
We can again use the cosine double angle formula in the above step, we get
\[\begin{align}
  & \Rightarrow 6{{\left( 2{{\cos }^{2}}x-1 \right)}^{2}}-3 \\
 & \Rightarrow 6\left( 2{{\cos }^{2}}x-1 \right)\left( 2{{\cos }^{2}}x-1 \right)-3 \\
\end{align}\]
We can now multiply the above terms we get,
\[\Rightarrow 6\left( 4{{\cos }^{4}}x-2{{\cos }^{2}}x-2{{\cos }^{2}}x+1 \right)-3\]
We can now simplify the above step, we get
\[\Rightarrow 6\left( 4{{\cos }^{4}}x-4{{\cos }^{2}}x+1 \right)-3\]
We can now multiply the terms inside the brackets, we get
\[\begin{align}
  & \Rightarrow 24{{\cos }^{4}}x-24{{\cos }^{2}}x+6-3 \\
 & \Rightarrow 24{{\cos }^{4}}x-24{{\cos }^{2}}x+3 \\
\end{align}\]
Therefore, the answer is \[24{{\cos }^{4}}x-24{{\cos }^{2}}x+3\].

Note: Students should know to split the term in order to use the cosine double angle formula. We should know some trigonometric identities like cosine double angle formula to solve these types of problems. In this problem we have used the formula twice and it can be used whenever required.