Question

How do you express $\cos \left( 4\theta \right)$ in the terms of $\cos \left( 2\theta \right)$ using the double angle identity?

Answer 1

This type of question is bases on the concept of integration. First, we have to consider the given function which can be expressed as $\cos \left( 2\times 2\theta \right)$. Assume $2\theta =x$ and thus we have to find the value of cos(2x). Use the identity of trigonometry, that is $\cos 2A={{\cos }^{2}}A-{{\sin }^{2}}A$, in the obtained expression. Then, we need to use the identity ${{\sin }^{2}}A+{{\cos }^{2}}A=1$ in the obtained equation and convert the sine function to cosine function. And then substitute $2\theta$ in terms of x to get the final required answer

Complete step by step solution:
According to the question, we are asked to express $\cos \left( 4\theta \right)$ in the terms of $\cos \left( 2\theta \right)$.
We have been given the function is $\cos \left( 4\theta \right)$. --------(1)
We can write the function (1) as
$\cos \left( 4\theta \right)=\cos \left( 2\times 2\theta \right)$.
Let us assume $2\theta$ to be x, that is $2\theta =x$.
Now, we get
$\Rightarrow \cos \left( 4\theta \right)=\cos \left( 2x \right)$.
We know that $\cos 2A={{\cos }^{2}}A-{{\sin }^{2}}A$.
Using this double angle identity of trigonometry, we get
$\cos 2x={{\cos }^{2}}x-{{\sin }^{2}}x$ ------------(2)
But we know that ${{\sin }^{2}}A+{{\cos }^{2}}A=1$.
We have to subtract ${{\cos }^{2}}A$ on both the sides of the identity.
$\Rightarrow {{\sin }^{2}}A+{{\cos }^{2}}A-{{\cos }^{2}}A=1-{{\cos }^{2}}A$
Since terms with opposite signs and same magnitude cancel out, we get
${{\sin }^{2}}A=1-{{\cos }^{2}}A$.
Using this property in the expression (2), we get
$\cos 2x={{\cos }^{2}}x-\left( 1-{{\cos }^{2}}x \right)$
$\Rightarrow \cos 2x={{\cos }^{2}}x-1+{{\cos }^{2}}x$
On further simplification, we get
$\cos 2x=2{{\cos }^{2}}x-1$.
But we have assumed $x=2\theta$.
On substituting the value of x in the above expression, we get
$\cos \left( 2\times 2\theta \right)=2{{\cos }^{2}}\left( 2\theta \right)-1$
Therefore, we get
$\cos \left( 4\theta \right)=2{{\cos }^{2}}\left( 2\theta \right)-1$
Hence, we can express $\cos \left( 4\theta \right)$ as $2{{\cos }^{2}}\left( 2\theta \right)-1$.

Note: We should be thorough with the identities of trigonometry to solve this type of questions. Avoid calculation mistakes based on sign conventions. Also we can solve this question without substituting $x=2\theta$. Instead we can solve the same keeping $2\theta$ as the angle.