1)Why is it that

sigma [ k=1 to (n-1) ] cos [2k(pi)/n] = -1 ????

2) Could you suggest any approach for this one ?

let n be an odd integer. If

sin [n(theta)] = sigma [ r=0 to n ] b (subscript r ) [sin (theta)]^r

then find the value of "b subscript 0 " and "b subsript 1 "

 hi,

consider the eq.

              x ^{n _{-1 = 0}}

^{_{this eq. will have n roots,  which will be the n th roots of unity,}}

   x ^{n _{= 1}}

x ^{n  _{=   e} i2¶k}

^{_{x =  e} i2¶k/n}

^{where k goes from 0  to n-1}

 ^{_{x=   1,  e} i2¶/n  _,e i4¶/n _,e i6¶/n _{.............................e} i2 (n-1)¶/n}

^{from the analysis of eq . we can tell that sum of all the roots will be 0 ,becoz the coeff. of  x^(n-10 is zero here}

^so,

 ^{_1+e i2¶/n  _+e i4¶/n _+e i6¶/n _{............................+.e} i2 (n-1)¶/n    =0}

^{_e i2¶/n  _+e i4¶/n _+e i6¶/n _{............................+.e} i2 (n-1)¶/n    _{= -1  hence proved}}

^_(ii)

 ^{_{sin nØ =  b0  + b1 sinØ  + b2  sin2Ø + b3  sin3Ø.................................+ bn sin nØ}}

^{_{in this case , we can find the values of b0 ,b1, b2..........  by succesive differentiation of both the}}

^{_{sides & then putting  Ø =0 , in that eq.}}

  ^{_{for b0,}}

  _{^{put Ø  = 0 , we get}}

_{^{b0 =0}}

_{^{for b1,}}

_{^{on diff. both the sides  & then putting Ø = 0, we get}}

 _{^{b1 = n}}

  _{^{for furthur coeff. we have to do furthur diff.}}