Question

Prove that cos2α = 1-tan²α / 1+tan²α​

Answer 1

see the explanation

Explanation:

we have

`cos(2\alpha)=(1-tan^2(\alpha))/(1+tan^2(\alpha))`

Prove

Remember that

`cos(2\alpha)=cos^2(\alpha)-sin^2(\alpha)`

`tan(\alpha)=(sin(\alpha))/(cos(\alpha))`

substitute in the expression above

`cos^2(\alpha)-sin^2(\alpha)=(1-((sin(\alpha))/(cos(\alpha)))^2)/(1+((sin(\alpha))/(cos(\alpha)))^2)`

`cos^2(\alpha)-sin^2(\alpha)=(((cos^2(\alpha)-sin^2(\alpha))/(cos^2(\alpha)))/(((cos^2(\alpha)+sin^2(\alpha))/(cos^2(\alpha)))}`

Remember that

`cos^2(\alpha)+sin^2(\alpha)=1`

Simplify

`cos^2(\alpha)-sin^2(\alpha)=cos^2(\alpha)-sin^2(\alpha)` ---> is proved