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Answer 1 · 2016-12-26T09:43:20

#sin(tan^-1u+sin^-1v)#

#=sin(cot^-1(1/u)+sin^-1v)#

#=sin(csc^-1sqrt(1+1/u^2)+sin^-1v)#

#=sin(csc^-1((sqrt(u^2+1))/u)+sin^-1v)#

#=sin(sin^-1(u/(sqrt(u^2+1)))+sin^-1v)#

#=sinsin^-1((u/(sqrt(u^2+1)))xxsqrt(1-v^2)+sqrt(1-u^2/(u^2+1))xxv)#

#=((usqrt(1-v^2))/(sqrt(u^2+1))+v/sqrt(u^2+1))#

#=(usqrt(1-v^2)+v)/sqrt(u^2+1)#

Answer 2 · 2016-12-26T10:54:02

#=(u sqrt(1-v^2)+v)/sqrt(1+u^2)#

Let #a = tan^(-1)u in Q_1, if u >= 0, and in Q_2 if u <= 0#.

#sin a = u/sqrt(1+u^2) and cos a= 1/sqrt(1+u^2)#

Let #b = sin^(-1)v in Q_1, if v >= 0, and in Q_2 if v <= 0#.

#sin b = v and cos v = sqrt(1-v^2)#

In any case, cos a and cos b > 0. sin a < 0, if u < 0.

The given expression is

#sin ( a + b ) = sin a cos b + cos a sin b#

#=(u/sqrt(1+u^2))(sqrt(1-v^2))+(1/sqrt(1+u^2))(v)#

#=(u sqrt(1-v^2)+v)/sqrt(1+u^2)#