Question

. Evaluate tan(α + β) = sin(????+????) / cos(????+????) to show tan(???? + ????) = tan(????)+tan(????????) / 1−tan(????) tan(????

. Use the resulting formula to show that
tan(2????) = 2 tan(????) / 1−tan2(????)

Answer 1

Answer with Step-by-step explanation:

We are given that

`tan(\alpha+\beta)`

`(sin(\alpha+\beta)/(cos(\alpha+\beta))`

By using formula
`tan x=(sin x)/(cos x)`

`(sin\alpha cos\beta+sin\beta cos\alpha)/(cos\alpha cos\beta-sin\alpha sin\beta)`

By using property:
`sin(x+y)=sin x cosy+cos x sin y`

`cos(x+y)=cos x cosy-sin x siny`

Divide numerator and denominator by
`cos\alpha cos\beta`

Then, we get

`((sin\alpha)/(cos\alpha)+(sin\beta)/(cos\beta))/(1-(sin\alpha sin\beta)/(cos\alpha cos\beta))`

`tan(\alpha+\beta)=(tan\alpha+tan\beta)/(1-tan\alpha tan\beta)`

Hence, proved

Substitute
`\alpha=\beta`

Then we get

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

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

Hence, proved.