68 BOOK OF MATHEMATICAL PROBLEMS. 



316. Prove that 



sin 2a sin (ft-y) + sin 2ft sin (y - a) -f sin 2y sin (a - ft) 

 = {sin (y - /?) + sin (a - y) 4- sin (/? - a)} 



{sin (ft + y) 4- sin (y 4- a) + sin (a 4- ft)}, 



and cos 2a sin (ft - y) 4- cos 2/? sin (y - a) + cos 2y sin (a - ft) 

 = {sin (y-ft) + sin (a - y) 4- sin ((3 - a)} 



{cos (ft 4- y) + cos (y 4- a) 4- cos (a + ft)}. 



317. Prove that 



sin 3a sin (ft - y) + sin 3ff sin (y - a) + sin 3y sin (a - ft) 

 ^ ' cos 3a sin (/3 - y) + cos 3/2 sin (y - a) + cos 3y sin (a - ft) 



= tan (a + ^3 + y), 



sin 5a sin (/? - y) + sin 5/? sin (y - a) + sin 5y sin (a - ft) 

 ^~' cos 5a sin (/i - y) + cos 5ft sin (y - a) + cos Oy sin (a - ft) 



sin 3a + 8 + + ... +... 



cos (3a + ft+y) +...+...' 

 and that 



sin 7a sin (/? y) + sin 7/? sin (y a) + sin 7y sin (a ^8) 

 ^ ' cos 7a sin (ft y) + cos 7/2 sin (y a) + cos 7y sin (a ft) 



_ sin (a + 3/2 + 3y) + ...+... + sin (5a 4- ft + y) + ...+... 

 ~~ cos (a + '6ft + 3y) +...+...+ cos (5a + ft + y) + ...+...' 



318. Prove that 



cos* a sin (B y) 4- cos 8 ft sin (y a) + cos 8 y sin (a ft) 

 sin* a sin (ft y)+ sin" /2 sin (y a) + sin 8 y sin (a ft) 



+ cot (a 4- ft + y) = 0. 



319. Prove that 



1 1 1 _sin?ia+asin(w l)a 



a- 2 cos a- 2 cos a- ...-2 cos a + a = sin(w+ l)a + a sinwa J 



there being n quotients in the left hand member. 



