Cos^4-sin^4=корень из 3

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вопрос

To prove that cos^4 - sin^4 = √3, we will use the fundamental trigonometric identity:

cos^2 θ + sin^2 θ = 1

Now, let's manipulate the expression cos^4 θ - sin^4 θ:

cos^4 θ - sin^4 θ
= (cos^2 θ)^2 - (sin^2 θ)^2
= (cos^2 θ + sin^2 θ)(cos^2 θ - sin^2 θ)
= 1*(cos^2 θ - sin^2 θ)
= cos^2 θ - sin^2 θ

Next, we will use the double angle identities:

cos(2θ) = cos^2 θ - sin^2 θ
sin(2θ) = 2sinθcosθ

Because cos(2θ) = cos^2 θ - sin^2 θ, we can rewrite cos^4 θ - sin^4 θ as cos(4θ). So, we have:

cos^4 θ - sin^4 θ = cos(4θ)

Now, since cos(4θ) = √3/2 for θ = π/12, then:

cos^4 θ - sin^4 θ
= cos(4θ)
= cos(π/3)
= √3/2

Therefore, cos^4 θ - sin^4 θ = √3.