Let #a = sin^(-1)(7x) in Q1 or Q4,# for the principal value..
Then, #sin a = 7x in [-1. 1], and so, x in[-1/7, 1/7].#
and #cos a =sqrt(1-49x^2) in [0, 1]#.
Let #b = tan^(-1)(7x) in Q1 or Q4#.
Then, #tan b =7x in[-1, 1] and sin b =(7x)/(1+49x^2)#.
#cos b = 1/sqrt(49x^2+1) in [0, 1]#.
Now, the given expressionis
#sin(b-a)#
#=sin b cos a - cos b sin a0#
#=(7x)/sqrt(1+49x^2)sqrt(1-49x^2)-(1/(sqrt(1+49x^2)))(7x)#
#=(7x)/sqrt(1+49x^2)(sqrt(1-49x^2)-1)#
Note that both sines and tangents of a and b are
#>=0#, when #x>=0#
and are #<=0#, when #x<=0#.
Cosines of of both a and b are #>=0#..
