Question

Solve the following equation
$\cos x+\cos 3x-2\cos 2x=0$

Answer 1

Hint : Simplify the given equation by using this identity, $\cos A+\cos B=2\cos \left( \dfrac{A+B}{2} \right)\cos \left( \dfrac{A-B}{2} \right)$ where we take A as 3x and B as x. After that, we use the identity that if $\cos x=\cos \alpha$ then the value of x is $2n\pi \pm \alpha .$

Complete step-by-step answer :
In this question, we are given an equation which is cos x + cos 3x – 2 cos 2x = 0 and we have to find the values of x for which the given equation satisfies. So, the given equation is,
$\cos x+\cos 3x-2\cos 2x=0$
Now, we will apply the following identity, which is,
$\cos A+\cos B=2\cos \left( \dfrac{A+B}{2} \right)\cos \left( \dfrac{A-B}{2} \right)$
Now, instead of values A and B, we will use 3x and x. So, we get,
$\Rightarrow \cos 3x+\cos x=2\cos \left( \dfrac{3x+x}{2} \right)\cos \left( \dfrac{3x-x}{2} \right)$
$\Rightarrow \cos 3x+\cos x=2\cos 2x\cos x$
So, we will apply this result in the following equation
$\cos x+\cos 3x-2\cos 2x=0$
$\Rightarrow 2\cos 2x\cos x-2\cos 2x=0$
Now, we will take 2 cos 2x common, we get,
\Rightarrow 2\cos 2x\left\\{ \cos x-1 \right\\}=0
So, the value of cos can either be 1 or cos 2x be 0 to find the value of x, we have to test the cases.
For cos x = 1, we can write it as, cos x = cos 0.
Now, we know that if, $\cos x=\cos \alpha ,$ then the value of x is $2n\pi \pm \alpha .$ So, we can write the value of x if cos x = cos 0 as $2n\pi \pm 0$ or $2n\pi$ where n belongs to integers.
Now, for cos 2x = 0, we can write it as, $\cos 2x=\cos \dfrac{\pi }{2}$
Now, we know that if, $\cos x=\cos \alpha ,$ then the value of x is $2n\pi \pm \alpha .$ So, we can write the value of 2x if $\cos 2x=\cos \dfrac{\pi }{2}$ as $2n\pi \pm \dfrac{\pi }{2}$ or x is equal to $n\pi \pm \dfrac{\pi }{4}$ where n is any integer.
Hence, the values of x are $2n\pi \text{ or }n\pi \pm \dfrac{\pi }{4}$ where n is any integer.

Note : We can also do by using the identities $\cos 2x=2{{\cos }^{2}}x-1$ and $\cos 3x=4{{\cos }^{3}}x-3\cos x$ and then substitute it to find the values of cos x. After this, we will apply the identity that if $\cos x=\cos \alpha$ then the value of x is $2n\pi \pm \alpha .$