Question

if `sin(alpha+beta)=1 and sin(alpha-beta)=1/2 " "0lealpha,beta,lepi/2,then` find `tan(alpha+2beta)and tan(2alpha+beta)`

Answer 1

`sin(alpha+beta)=1`
`alpha+beta=pi/2-(1)`
`sin(alpha-beta)=1/2`
`alpha-beta=1/2`
`alpha-beta=pi/6-(2)`
`alpha+beta=pi/2`
from equation 1 and 2
`alpha=pi/3`
`alpha+beta=pi/2`
`beta=pi/2-pi/3=pi/6`
`tan(pi/3+2*pi/6)=tan(pi/3+pi/3)=taan2/3pi=-sqrt3`
`tan(2alpha+beta)=tan(2*pi/3+pi/6)=tan(2/3pi+pi/6)`
`=tan((4pi+pi)/6)`
`tan5/6pi=tan(pi-pi/6)`
`-tanpi/6=-1/sqrt3`.