64 BOOK OF MATHEMATICAL PROBLEMS. 



297. If cos A + cos B cos C + sin B sin C cos A = 0, 



then will tan ^45 * ^\ tan ^45 - |) tan ^45 - ^ = - 1 ; 



and if cos A = cos B cos C sin B sin (7 cos A, 



then will cos B = cosC cos -4 sin C sin J. cos j5. 



298. Reduce to its simplest form the equation 

 {x cos (a + /3) + y sin (a + /?) - cos (a - /?)} 



{# cos (y + 8) + ?/ sin (y + 8) cos (y 8)} 



= {x COS (a + y) + y sin (a + y) - COS (a - y)} 



{x cos ( + 8) + y sin (0 + S) - cos (/? - 8)}. 



299. Given the equations 



yz-y'z + zx -zx + xy'-x'y = Q, A + JB + C= 180, 

 xx 1 sin s A + yy sin 2 5 + zz sin 2 (7 = (yz + y'z) sin 2? sin C cos -4 



+ (zx' + s'ce) sin (7 sin A cos jB + (xy + x'y) sin -4 sin B cos (7; 

 prove that, for real values, either x = y = z ; or x = y = z'. 



300. If A + B + C = 180, and if 



y* + z' + 2yz cos A _ z* + x* + 2zx cos B _ x* + y* + 2xy cos C 

 ~rinM~ sin 2 ^ sin 3 ^ 



then will either 



x sin A + y sin B + z sin C = 0, or -. = r> = ^ . 



cos A cos B cos (7 



301. Having given the equations 



U~ + z* 2 7/2 cos a 2 + x* 2zx cos /3 x* + y 3 2xy cos y 

 bin" a sin 8 /? sin 2 y 



