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.

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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**

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