The most straightforward way to obtain the expression for cos(2*x*) is by using the "cosine of the sum" formula: cos(x + y) = cosx*cosy - sinx*siny.

To get cos(2*x*), write 2x = x + x. Then,

cos(2*x*) = cos(*x* +* x*) = cosx*cosx - sinx*sin*x* = cos² (*x*) - sin² (x)

Then, from Pythagorean Identity, we can get express sine in terms of cosine:

sin² (*x*) + cos² (*x*) = 1

sin² (*x*) = 1 - cos² (*x*)

Plugging this into the formula for cos(2x), we get

cos(2*x*) = cos² (*x*) - (1 - cos²(*x*)) = 2cos²(x) - 1.

Alternatively, we could get the expression for cos(2*x*) in terms of sin(x):

cos(2*x*) = (1 - sin²(*x*)) - sin²(*x*) = 1 - 2sin²(x)

So, there are three formulas for the cosine of the double angle:

**cos(2 x) = cos²(x) - sin²(x) **

** = 1 - 2sin²(x) **

** = 2cos²(x) - 1.**

The last one writes the cosine of 2*x *in terms of cosine of x.

We know that the expression for cos ( x + y) is:

cos (x + y) = cos x * cos y - sin x* sin y

Now substituting x for both x and y we get

cos ( x + x) = cos x * cos x - sin x * sin x

=> cos 2x = (cos x)² - (sin x)²

Now we use the relation (cos x)² + (sin x)² = 1 which gives (sin x)² = 1 - (cos x)²

So we can eliminate (sin x)² and get

=> cos 2x = (cos x)² - 1 + (cos x)²

=> cos 2x = 2*( cos x)² - 1

Therefore in terms of cos x , **cos 2x = 2*( cos x)² - 1**

cos2x = cos (x+x)

cosxcosx - sinxsinx

(cosx)^2-(sinx)^2

Since (sinx)^2 = 1-(cosx)^2

Therefore: = (cosx)^2 -(1-(cosx)^2)

= (cosx)^2 -1+(cosx)^2

** = 2(cosx)^2 -1**

We'll write cos 2x as the cosine of the sum of 2 like angles:

cos(x+x) = cos x*cos x - sin x*sin x

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

We'll write sin x in terms of cos x, applyingthe fundamental formula of trigonometry:

(sin x)^2 + (cosx)^2 = 1

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

We'll substitute (2) in (1):

cos(x+x) = (cos x)^2 - [1 - (cos x)^2]]

We'll remove the brackets:

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

We'll combine like terms:

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

So,the expression of cos 2x, written in terms of cos x, is:

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