Question

How do you prove that `1- cos^2(x/2) = (sin^2x)/(2(1+cosx))`?

Answer 1

Using that
`cos(x)=2cos^2(x/2)-1`

Explanation:

so we have
`2(1+cos(x))=2(2cos^2(x/2)-1)=4cos^2(x/2)`
so we get
`4cos^2(x/2)(1-cos^2(x/2))`
and
`sin^2(x)=1-cos^2(x)`
`1-(2cos^2(x/2)-1)^2`
`1-4cos^2(x/2)-1+4cos^2(x/2)`
`4cos^2(x/2)(1-cos^2(x/2))`

Answer 2

Please find a Proof in Explanation.

Explanation:

Since, `1-cosx=2sin^2(x/2)`, we have,

`sin^2x/(2(1+cosx))=(1-cos^2x)/(2(1+cosx))`,

`={cancel((1+cosx))(1-cosx)}/(2cancel((1+cosx))`,

`=(1-cosx)/2`,

`={cancel(2)sin^2(x/2)}/cancel(2)`,

`=sin^2(x/2)`,

`=1-cos^2(x/2)`, as desired!