70 BOOK OF MATHEMATICAL PROBLEMS. 



325. Having given the equation 



sec ft secy + tan ft tan y = tan a, 

 prove that, for real values of ft and y, cos 2a must be negative ; 



and that 



tan ft + tan a tan y cos ft 

 tan y + tan a tan ft cos y 



326. If A + B + C= 180, prove that 



A J? P 



sin -TT sin -^ sin ^r > (1 cos -4) (1 cos B)(l cos C) 



J A J 



> cos ^1 cos -5 cos (7, 

 unless A = B = C. Also 



sin -4 sin .5 sin (7 > sin 2 A sin 25 sin 2(7. 



327. Prove that 



/cos 2 (a - 0) sin 8 (a - 6)) ( cos 2 (a + 0) sin 2 (a + 0)) sin 2 2a 

 I 1^ ~b A ^~ "i 5 r V&* ; 



and that the two cannot be equal unless tan 2 a lie between 



6 f ,a f 



~s and .... 

 a* b* 



328. If tan a tan ft tan y = 1, a, ft, y being angles between 

 and |, sin a sin0 siny < j , 



unless a = ft = y. 



329. If a + A, ft + B, y + C be the angles subtended at any 

 point by the sides of a triangle ABC, 



sin* a sin*/? sin a y sin a sin ft sin y 



sin ,4 sin .# + sin G * ^ . A . B . C ' 

 2 sin sin sin 



A A A 



unless the point be the centre of the inscribed circle. 



