Question

(iii) cos 4x = cos 2x​

Answer 1

For this, we may know that, for two angles A and B,

cos A = cos B   ⇒   A = 2nπ ± B,   n ∈ Z.

Here, we're given,

              cos(4x) = cos(2x​)

According to the formula, we get two cases,

Case 1 :-  4x = 2nπ ± 2x

⇒            4x = 2(nπ ± x)

⇒            2x = nπ ± x

⇒            x = nπ          OR          x = nπ / 3

Case 2 :-  2x = 2nπ ± 4x

⇒            2x = 2(nπ ± 2x)

⇒            x = nπ ± 2x

⇒            x = -nπ          OR          x = nπ / 3

Since n ∈ Z, it doesn't matter x equals nπ or - nπ, because both situation contain all possible values of n.

Answer 2

Solution :

• Given equation is cos 4x = cos 2x

$\implies\sf \: 4x = 2n\pi +or - 2x \\ \implies\sf \: 4x = 2n\pi + 2x \: or \: \\ \implies\sf \: 4x = 2n\pi - 2x \\ \implies\sf \: 2x = 2n\pi \: or \: \\ \implies\sf \: 6x = 2n\pi$

$\implies\sf \: x = n\pi$

or

$\implies\sf \: x = \frac{n\pi}{3} ,\: n \in \: Z$

$\implies\bf \: ie \: x = \frac{n\pi}{3} \: or \: x = n\pi, \: n\in \: Z$