Question

Which of the following expressions is equal to 1 – cos4 θ?

Answer 1

`1-\cos^4\left(θ\right)=(1-\cos^2\left(θ\right))(1+\cos^2\left(θ\right))`

Given that: sin^2 θ = 1 - cos^2 θ, then

`1-\cos^4\left(θ\right)=(1+\cos^2\left(θ\right))\sin^2\left(θ\right)=\sin^2\left(θ\right)+\sin^2\left(θ\right)\cos^2\left(θ\right)`
`1-\cos^4\left(θ\right)=(1+\cos^2\left(θ\right))\sin^2\left(θ\right)=\sin^2\left(θ\right)+\sin^2\lparenθ)(1-\sin^2\lparenθ))`
`1-\cos^4\left(θ\right)=\sin^2\left(θ\right)+\sin^2\lparenθ)(1-\sin^2\lparenθ))=\sin^2\left(θ\right)+\sin^2\lparenθ)-\sin^4\lparenθ)=2\sin^2-\sin^4\lparenθ)`
`1-\cos^4\left(θ\right)=2\sin^2-\sin^4\lparenθ)`

So the answer is the first option.