56 DIFFERENTIAL AND INTEGRAL CALCULUS. 



3 



cot a:. - 



14 



2. From the equality 



sinx sin (a + x] sin(2a + a?) ... sin {(ft 1) a 4 x} = 2 1 ~ H sinrca?, 

 where no. = TT, 



find, by taking the logarithmic differential coefficient of 

 both members, the sums of the series 



(1) cotx + cot (a 4 x) 4 cot (2a 4- a?)...+ cot {(n - 1) a + a;}, 



3. Having given that 



x x x s'm2x 



cosa? cos- cos-^ ... cos -^j ... to co - 



22 2 '2.I 4 



prove that 



x I x 

 (1) tanaj-f | tan- + -, tan -,+... 



1 x 1 



=! tan o'-i + -" to GO = - -2 cot2ic, 



& M tC 



(2) 



4 -pr5 tan 2 -f-, +. . . to oc = - -5 + 4 cot"2. 



i 4: Z X 



4. Find the differential coefficients of 



x x -* 



- X 



a vers-'g) + V(^-^), tan^g) 



2 . I a - 



, _ _ _ , 



a 2 ^ 



- a tan 



