Prove that the cosine of double angle may be written in terms of sine of angle.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll write the cosine of double angle as cos 2x. We know that the cosine of double angle can be written as 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 cos x in terms of sin x, applying the fundamental formula of trigonometry:

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

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

We'll combine like terms:

cos 2x = 1 - 2(sin x)^2

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

cos 2x = 1 - 2(sin x)^2

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