Question

Prove the following equality Cosx⁴ - Sinx⁴ = Cos2x

Answer 1

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.