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Let sinx + siny = a and cosx + cosy = b. Then sin(x+y) is equal to a) ab/(a+b) b) ab/(a 2 +b 2 ) c) 2ab/(a 2 +b 2 ) d) ab/2(a 2 +b 2 )
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 option c is correct
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
option c is correct
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