BOOK OF MATHEMATICAL PROBLEMS. 



VIII. Miscellaneous Questions. 



401. The ambiguities in the equations 



A A A A 



cos - 9 - + sin jr- = ^/(l + sin A), cos -^ - sin _ - = ^/(l - sin A) 



may be replaced by ( 1)"*, ( 1)", where m, n denote the greatest 



. ^ + 90 . A + 270 

 integers in 36QO , and 36QU respectively. 



402. If a + 1 (tan + tan <) + c tan tan <j> = 0, prove that 



1 1 



a cos 8 + 26 sin cos + c sin 8 a cos 8 $ -f 26 sin </> cos <j> + c sin s </> 



a+c 



is equal to iT . 



c - 6 



403. AB is a fixed straight line, C a point in it such that 

 SAC = AB, and P a point between B and C : if CP = CB sin 0, 

 then will .5P . AP* cc 1 + sin 30. Hence prove that BP . AP* is 

 greatest when P bisects BC. 



404. Find x from the equation 



1 



and find the tangent of the angle 



405. Three parallel straight lines are drawn through the 

 angular points of a triangle ABC to meet the opposite sides in 

 Z>, E, F- t prove that 



DB.DC EC.EA FA.FB 



J)A* B FC 



the segments of a side being affected with opposite signs when 

 they fall on opposite sides of the point of section. 



