56 EXAMPLES OF DIFFERENTIAL COEFFICIENTS. 



. n + 1 . nx 

 sin x sin 



78. Given sina; + sin2a; + ... +smnx= , 



. x 

 sin- 



deduce, by taking the differential coefficients of both sides, 

 the sum of 



cosoj + 2 cos2a;+ +n cosnx. 



n + 1 . x . 2n + l 1 / . n + 1 \ 2 

 sm - sin 



Eesult. ' 



. X 



sin 

 2 



79. Having given (see Plane Trigonometry, Chap, xxm.) 



. fir \ . /2-7T \ . fm 1 \ sin mx 



sma;sm \-x\ sm ( \- x) ... sin TT + X) = 



\m ) \m J \ m ) 



where m is a positive integer, shew that 



cot x -f- cot ( h x ] + ...+ cot f TT + x } = m cot mx. 



\m J \ m J 



80. From the preceding result deduce that 



(*rr \ /iwt 1 N 

 Ha;] + ... + cosec 2 [ TT + X) 

 m J \ m J 



m cosec mx. 



