cos a +cos b =1/3 and sin a +sin b=1/4 prove that tan(1/2(a+b))=3/4

cosa+cosb=1/3

2cos(a+b/2)cos(a-b/2) = 1/3  ...............  (1)

sina+sinb = 1/4

2sin(a+b/2)cos(a-b/2) = 1/4 ................... (2)

(1)/(2)  we get

tan(a+b/2) = 1/3*4/1

tan(1/2(a+b) = 3/4