how to write cos 2x in terms of cos x?

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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**

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**

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