Use the sine angle addition formula:
#sin(color(red)x+color(blue)y)=sincolor(red)xcoscolor(blue)y+coscolor(red)xsincolor(blue)y#
To simplify the identity, use the above formula, split up the fraction
#color(white)=sin(color(red)alpha+color(blue)beta)/(cosalphacosbeta)#
#=(sincolor(red)alphacoscolor(blue)beta+coscolor(red)alphasincolor(blue)beta)/(cosalphacosbeta)#
#=(sincolor(red)alphacoscolor(blue)beta)/(cosalphacosbeta)+(coscolor(red)alphasincolor(blue)beta)/(cosalphacosbeta)#
#=(sincolor(red)alphacolor(red)cancelcolor(black)(coscolor(blue)beta))/(cosalphacolor(red)cancelcolor(black)(cosbeta))+(color(red)cancelcolor(black)(coscolor(red)alpha)sincolor(blue)beta)/(color(red)cancelcolor(black)(cosalpha)cosbeta)#
#=sincolor(red)alpha/cosalpha+sincolor(blue)beta/cosbeta#
#=tanalpha+sincolor(blue)beta/cosbeta#
#=tanalpha+tanbeta#
