prove the identity cos 3 cos7 sin 7 sin3 tan2SolutionThe g

prove the identity

cos 3 - cos7 / sin 7 +sin3 =tan2

Solution

The given

cos 3 - cos7 / sin 7 +sin3 =tan2

Let LHS=

cos 3 - cos7 / sin 7 +sin3

{-2sin((10)/2)sin((-4)/2)}/2sin((10)/2)cos((4)/2) (since cos c-cos d=-2sin((c+d)/2)sin((c-d)/2) and

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

LHS={-2sin((10)/2)sin((-4)/2)}/2sin((10)/2)cos((4)/2)

=-sin((-4)/2)/cos((4)/2) by simplification

=sin2/cos2 (since sin(-2)=-sin2)

  =tan2

therefore LHS=RHS