Let S = x (- , ) x 0, 2 . The sum of all distinct solutions of the equation 3 x+ cosec x+2( x- x)=0 in the set S is equal to

Step-by-step solution

aligned & 3 x+ cosec x+2( x- x)=0 & 3 2 x+ 1 2 cosec x= x- x & 3 2 1 x + 1 2 1 x = x x - x x & 3 2 x+ 1 2 x= ^2 x- ^2 x & ( 3 -x )= 2 x & 3 -x=2 n 2 x aligned for n =0: x= 9 (- , ) or - 3 (- , ) for h =1: x= -5 9 (- , ) or x= 5 3 (- , ) for n =-1: x= -7 3 (- , ) or x= 7 9 (- , ) for n=2: x= 11 3 (- , ) or x= -11 9 (- , ) Distinct solutions are : 9 , - 3 , -5 9 , 7 9 Required sum = -3 -5 +7 9 =0