Question

What is difference between `2cosx` and `cos2x`?

As example:
`cos3x= cos2x*cosx` Why?
`2cos3x=?`

But how:
`2cos2x+2cos2x=?`
`2cos2x*2cos2x=?`

Could you explain and prove? How to avoid misunderstanding when solving complex equations? It's easy to just use the formula in simple cases

Answer 1

`2cos x - cos 2x = -2cos^2x + 2 cos x - 1`

Explanation:

They're very different and your example is not correct.

`2 cos x` is twice the cosine of angle `x`. It will be between `-2` and `2.`

`cos2x` is an abbreviation for `cos(2x).` It is the cosine of the angle `2x`, two times the angle `x`. The value of `cos 2x` will be between `-1` and `1.`

Before we think about double angles, let's remember the formula for the cosine of the sum of two angles:

`cos(a+b) = cos a cos b - sin a sin b`

We substitute `a=b=x.`

`cos 2x = cos(x + x) = cos x cos x - sin x sin x = cos ^2 x - sin ^2 x`

Since `cos^2 x + sin ^2x = 1` we can also write

`cos 2 x = cos^2x - (1- cos^2x) = 2 cos^2 x - 1`

and

`cos 2x = (1 - sin ^2 x) - sin^2 x = 1 - 2 sin ^2 x`

So there are (at least) three double angle formulas for cosine:

`cos 2x = cos ^2 x - sin ^2 x = 1 - 2 sin ^2 x = 2 cos^2 x - 1`

The one with just cosine on the right is usually the most useful.

The example was `cos 3x.` Let's work out the triple angle formula for cosine.

`cos 3x = cos(x + 2x) = cos x cos 2 x - sin x sin 2x`

We need the double angle formula for sine, which is

`sin 2x = 2 sin x cos x`

`cos 3x = cos x (2 cos ^2 x -1) - sin x (2 sin x cos x)`

`cos 3x = cos x (2 cos ^2x - 1 - 2 sin^2 x )`

`cos 3x = cos x (2 cos ^2x - 1 - 2 (1 - cos^2 x) )`

`cos 3x = cos x (4 cos ^2x - 3 )`

`cos 3x = 4 cos ^2 x - 3 cos x`

Let's literally answer the question:

`2cos x - cos 2x = 2 cos x - (2 cos^2 x - 1) = -2cos^2x + 2 cos x - 1`