1.if a+b+c=180degrees,prove that sina + sinb + sinc=4cosa/2cosb/2cosc/2 2.Prove That sin8theta costheta - cos3theta sin6theta/cos2theta costheta - sin3thetasin4theta =tan2theta 3.Find perpendicular distance of the line joining the points(cos theta,sin theta)and (cos phy,sin phy)from the origin.

We would request you to please ask one question at a time. The solution to your first question is as follows:

 

sinA + sinB + sinC = 2sin[(A+B)/2]cos[(A-B)/2] + 2sinC/2cosC/2

                                  = 2sin[(?-C)/2]cos[(A-B)/2] + 2sinC/2cosC/2

                                  = 2cosC/2cos[(A-B)/2] + 2sinC/2cosC/2

                                  = 2cosC/2* [cos[(A-B)/2]+[cos(A+B)/2]          [Since, sinC/2 = cos(A+B)/2]

Now use cosC + cosD = 2 cos[(C+D)/2]cos[(C-D)/2]

                                  = 2cosC/2* [2cosA/2 cosB/2]

                                  = 4 cosA/2 cosB/2 cosC/2