Question

How i prove this ?

Answer 1

Please see below.

Explanation:

As `sin(alpha+-beta)=sinalphacosbeta+-cosalphasinbeta`

and `cos(alpha+-beta)=cosalphacosbeta∓sinalphasinbeta`

Hence `(sin(alpha+beta)-2sinalphacosbeta)/(2sinalphasinbeta+cos(alpha+beta))`

= `(sinalphacosbeta+cosalphasinbeta-2sinalphacosbeta)/(2sinalphasinbeta+cosalphacosbeta-sinalphasinbeta)`

= `(sinbetacosalpha-cosbetasinalpha)/(cosbetacosalpha+sinbetasinalpha)`

= `sin(beta-alpha)/cos(beta-alpha)`

= `tan(alpha-beta)`