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

Hint: 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.