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

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

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

cos(2x) = cos(x + x) = cosx*cosx - sinx*sinx = 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(2x) = cos² (x) - (1 - cos²(x)) = 2cos²(x) - 1.

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

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

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

cos(2x) = cos²(x) - sin²(x)

= 1 - 2sin²(x)

= 2cos²(x) - 1.

The last one writes the cosine of 2x 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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