How do you simplify #sin(tan^-1(7x)-sin^-1(7x))#?

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Answer 1 · 2016-08-01T09:47:23

#=(7x)/sqrt(1+49x^2)(sqrt(1-49x^2)-1), x in [-1/7, 1/7]#

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#..