Question

Find the value of $m$ from the cosine polynomial expression $\cos 6x=32{{\cos }^{6}}x-48{{\cos }^{4}}x+18{{\cos }^{2}}x-m$.

Answer 1

Hint: Expand $\cos 6x$ using the trigonometric formula of $\cos 2\theta $. During expansion also use the trigonometric formulas of $\cos 3\theta $ and algebraic identities. Comparing the constant term in the final expression with the constant term of given cosine expression will give you the value of $m$.

Complete step-by-step answer:
The given equation is \[\]
$\cos 6x=32{{\cos }^{6}}x-48{{\cos }^{4}}x+18{{\cos }^{2}}x-m....(1)$ \[\]
We know from the trigonometric identity,
\[\cos 2\theta =2{{\cos }^{2}}\theta -1\]
We shall use the above trigonometric identity to expand $\cos 6x$. Therefore,\[\]
$\cos 6x=\cos 2\left( 3x \right)=2\cos {{\left( 3x \right)}^{2}}-1$\[\]
Now again we shall use the trigonometric identity$\cos 3\theta =4{{\cos }^{3}}\theta -3\cos \theta $ and the algebraic identity ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$ we proceed,\[\]
$\begin{align}
  & \cos 6x=2{{\left( 4{{\cos }^{3}}x-3\cos x \right)}^{2}}-1 \\
 & =2\left[ 16{{\cos }^{6}}x+9{{\cos }^{2}}x-24{{\cos }^{4}}x \right]-1 \\
 & =32{{\cos }^{6}}x-48{{\cos }^{4}}x+18{{\cos }^{2}}x-1 \\
\end{align}$ \[\]
Comparing constant term in the above equation with constant term in the equation (1) we get,
\[-1=-m\Rightarrow m=1\]
So the value of $m$ is 1.\[\]
Alternate method:\[\]
We substitute an appropriate value in the equation (1) . The domain cosine function is all real numbers. Therefore we can take $x=0$ which lies in the domain of $\cos x$ and put it in the equation (1). The equation transforms to ,\[\]
$\begin{align}
  & \cos 6\cdot 0=32{{\cos }^{6}}0-48{{\cos }^{4}}0+18{{\cos }^{2}}0-m \\
 & \Rightarrow 1=32-48+18-m \\
 & \Rightarrow 1=2-m \\
 & \Rightarrow m=1 \\
\end{align}$ \[\]
So the value of $m$ is 1.

Note: The question tests your knowledge in application of different trigonometric identities and formulas. We shall be careful to use them with proper substitution because the incorrect substitution will lead to incorrect results. The alternate method will consume less time than the first method and that should be used during entrance examinations. The substitution $x=0$ is the shortest way to find out the value of $m$.