Mastering The Double Angle Identity For Cosine And Its Sin Equivalent

Double Angle Identity for Cosine II (sin)

cos2x=1-2sin^2(x)

The double angle identity for cosine states that:

cos(2θ) = 1 – 2sin^2(θ)

To rewrite this identity using sin, we can start with the Pythagorean identity for sine and cosine:

sin^2(θ) + cos^2(θ) = 1

Solving for cos^2(θ), we get:

cos^2(θ) = 1 – sin^2(θ)

Now we can substitute this into the double angle identity for cosine:

cos(2θ) = 1 – 2sin^2(θ)

cos(2θ) = cos^2(θ) – sin^2(θ)

cos(2θ) = (1 – sin^2(θ)) – sin^2(θ)

cos(2θ) = 1 – 2sin^2(θ)

Therefore, the double angle identity for cosine can also be expressed using sine as:

cos(2θ) = 1 – 2sin^2(θ)

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