Question

sinx+siny=a & cosx+cosy=b
then tan²(x+y)/2+tan²(x-y)/2 =?

Answer 1

sinx + siny = a      

2sin(x+y/2)cos(x-y/2) = a            ......................1                   (by using sum formula)

 

cosx + cosy = b

2cos(x-y/2)cos(x+y/2) = b                    ..................2

 

divide both

tan(x+y/2) = a/b               ...........3

tan2@ = sec2@ - 1           &         sec@ = 1/cos@ so

cos(x+y/2) = [b2/a2+b2]1/2 ......................4

sin(x+y/2) = {1-cos2(x+y/2)}1/2 = [a2/a2+b2]1/2.............5

 

sinx+y = 2sin(x+y/2)cos(x+y/2)                                    (sinA = 2sinA/2cosA/2)

from eq 4 & 5

sinx+y = 2ab/a2+b2