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Prove the following equality Cosx⁴ - Sinx⁴ = Cos2x

User Teucer
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Answer:

To prove:

Cos⁴x - Sin⁴x = Cos2x​

we get,

Consider, (Left hand side (LHS))


\cos^4x-\sin^4x

we know that,


(a+b)(a-b)=a^2-b^2

Usind this formula we get,


\cos^4x-\sin^4x=(\cos^2x)^2-(\sin^2x)^2
=(\cos^2x+\sin^2x)(\cos^2x-\sin^2x)

we know that,


\begin{gathered} \cos2x=\cos^2x-\sin^2x \\ \cos^2x+\sin^2x=1 \end{gathered}

Substitute the values we get,


=(1)(\cos2x)
\cos^4x-\sin^4x=\cos2x

Hence proved.

User Rosaura
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