Question

prove the identity

(sinb+sinc)/(cosb+cosc)=tan(b+c/2)

Answer 1

(sinb+sinc)/(cosb+cosc)

 

Recall the sum and product rules of trigonometry

sin b+ sin c=2sin((b+c)/2)cos((b-c)/2)

cos b + cos c=2 cos ((b+c)/2) cos ((b-c)/2)

 

(sinb+sinc)/(cosb+cosc)

    =2sin((b+c)/2)cos((b-c)/2)/2 cos ((b+c)/2) cos ((b-c)/2)

    =sin((b+c)/2)/cos ((b+c)/2)                      (Cancel the common terms)

    =tan(b+c/2)