80 BOOK OF MATHEMATICAL PROBLEMS. 



VI. Expansions of Trigonometrical Functions. 



380. By means of the equivalence of the expansions of 



2c* sin x x e* cos x, and e 2 * sin 2#; 



prove that 



.TT mr nn , . mr 

 sin (n r) T cos r 2 sin 



>r = n-l 4 4: 4 



381. Prove by comparing the coefficients of 2 "" 1 that the 

 expansions of sin and cos in terms of satisfy the equality 



2 sin cos = sin 20. 



382. Prove that 



i n ( n + 1) T 



... / x ,, 

 (ring 



383. From the expansion of (sin 0) 2n+1 in terms of sines of 

 multiples of 0, prove that 



to (n + 1) terms. 



384. If n be an odd integer, 

 Ti1 . n I 2n 1 



1+- -C080 + - - COS 20+. ..to 00 



n n 2n 



& cos n^ 



Within what limits of is this true 1 



