How do you use the double angle or half angle formulas to derive $ \cos (4x) $ in terms of $ \cos x $ .

Hint: Here this question is related to the topic trigonometry, the cosine is the trigonometry ratio. Now here we want to find the $ \cos (4x) $ in terms of $ \cos (x) $ . To solve this, we use the double angle formula or half angle formula. Hence, we can obtain the solution for this question.

Complete step-by-step answer:
To solve the above equation we use a double angle formula. Formulas expressing trigonometric functions of an angle $ 2x $ in terms of functions of angle $ x $ .
In general, the double angle for $ \cos (2x) $ is defined as
$ \cos (2x) = 2{\cos ^2}x - 1 $
The question can be written as $ \cos (2(2x)) $ , we apply the double angle for this we get
 $ \Rightarrow \cos (2(2x)) = 2{(\cos (2x))^2} - 1 $
Now again we use the double angle formula for cosine we get
 $ \Rightarrow \cos (4x) = 2{(2{\cos ^2}x - 1)^2} - 1 $
To solve further we use the standard algebraic formula $ {(a - b)^2} = {a^2} - 2ab + {b^2} $ . Here $ a = 2{\cos ^2}x $ and b=1.
Therefore, by applying the formula we get
 $ \Rightarrow \cos (4x) = 2(4{\cos ^4}x + 1 - 4{\cos ^2}x) - 1 $
On simplification
 $ \Rightarrow \cos (4x) = 8{\cos ^4}x - 8{\cos ^2}x + 2 - 1 $
On further simplification we get
 $ \Rightarrow \cos (4x) = 8{\cos ^4}x - 8{\cos ^2}x + 1 $
Hence, we have solved the $ \cos (4x) $ in terms of $ \cos (x) $ by using the double angle formula. The solution involves the cosine terms. In the solution we have cosine terms as fourth power and square of cosine.
We can also answer this question by using the half angle formulas. The half angle formula is defined as $ \cos \left( {\dfrac{x}{2}} \right) = \pm \sqrt {\dfrac{{1 + \cos x}}{2}} $ and on further we can use double angle formula or trigonometry identities to solve further.
So, the correct answer is “ $ \cos (4x) = 8{\cos ^4}x - 8{\cos ^2}x + 1 $”.

Note: For the trigonometry ratios we have double angle formula and half angle formula. By using these formulas, we can solve the trigonometry ratios. The double angle formula for cosine is defined as $ \cos (2x) = 2{\cos ^2}x - 1 $ and the half angle formula is defined as $ \cos \left( {\dfrac{x}{2}} \right) = \pm \sqrt {\dfrac{{1 + \cos x}}{2}} $ where x represents the angle. Hence we can obtain the required solution for the question.