Question

How do you prove $\cos 4x = 8{\cos ^4}x - 8{\cos ^2}x + 1$?

Answer 1

This question is related to the trigonometry, and we have to prove that the left hand side is equal to the right hand side of the expression, and this question can be solved by using trigonometric identities i.e.,$\cos 2x = 2{\cos ^2}x - 1$, and using the algebraic identity${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ and by simplifying the expression further we will get the result which is on the right hand.

Complete step-by-step solution:
Given $\cos 4x = 8{\cos ^4}x - 8{\cos ^2}x + 1$,
Now we have to prove that left hand side is equal to the right hand side of the equation, now take the expression on the left hand side i.e.,
$\Rightarrow \cos 4x$,
This can be rewritten as $\cos 4x = \cos 2(2x)$,
Now using the trigonometric identity i.e., $\cos 2x = 2{\cos ^2}x - 1$,
Now substituting the identities in the left hand side of the given expression we get, here we have $2x$ in place of $x$, so the identity can be written as,
$\Rightarrow \cos 4x = \cos 2(2x) = 2{\cos ^2}2x - 1$,
Now again using the trigonometric identity i.e., $\cos 2x = 2{\cos ^2}x - 1$,
$\Rightarrow \cos 4x = 2{\left( {2{{\cos }^2}x - 1} \right)^2} - 1$,
Now using the algebraic identity${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$,
$\Rightarrow \cos 4x = 2\left( {{{\left( {2{{\cos }^2}x} \right)}^2} - 2\left( {2{{\cos }^2}x} \right)\left( 1 \right) + {1^2}} \right) - 1$,
Now simplifying we get,
$\Rightarrow \cos 4x = 2\left( {4{{\cos }^4}x + 1 - 4{{\cos }^2}x} \right) - 1$,
Now taking out the brackets we get,
$\Rightarrow \cos 4x = 8{\cos ^4}x + 2 - 8{\cos ^2}x - 1$,
Now simplifying further we get,
$\Rightarrow \cos 4x = 8{\cos ^4}x - 8{\cos ^2}x + 1$
Which is equal to the right hand side of the equation,
Hence proved.

$\therefore$By using identities we proved that expression $\cos 4x$ is equal to $8{\cos ^4}x - 8{\cos ^2}x + 1$.

Note: An identity is an equation that always holds true. A trigonometric identity is an identity that contains trigonometric functions and holds true for all right-angled triangles. They are useful when solving questions with trigonometric functions and expressions. There are many trigonometric identities, here are some useful identities:
${\sin ^2}x = 1 - {\cos ^2}x$,
${\cos ^2}x + {\sin ^2}x = 1$,
${\sec ^2}x - {\tan ^2}x = 1$,
${\csc ^2}x = 1 + {\cot ^2}x$.
${\cos ^2}x - {\sin ^2}x = 1 - 2{\sin ^2}x$,
${\cos ^2}x - {\sin ^2}x = 2{\cos ^2}x - 1$,
$\sin 2x = 2\sin x\cos x$,
$2{\cos ^2}x = 1 + \cos 2x$,
$\tan 2x = \dfrac{{2\tan x}}{{1 - {{\tan }^2}x}}$.