Double Angle Identity for Cosine I (sin & cos)
cosine2x=cos^2(x)-sin^2(x)
The double angle identity for cosine states that:
cos(2θ) = cos²(θ) – sin²(θ)
or
cos(2θ) = 2cos²(θ) – 1
We can derive this identity using the angle addition formula for cosine:
cos(α + β) = cos(α)cos(β) – sin(α)sin(β)
If we let α = β = θ, we get:
cos(2θ) = cos(θ + θ) = cos(θ)cos(θ) – sin(θ)sin(θ)
which simplifies to:
cos(2θ) = cos²(θ) – sin²(θ)
We can also use the Pythagorean identity (sin²(θ) + cos²(θ) = 1) to rewrite this as:
cos(2θ) = cos²(θ) – (1 – cos²(θ))
which simplifies to:
cos(2θ) = 2cos²(θ) – 1
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