Show that? : # tan(arcsinx) = x/sqrt(1 -x^2) #
1 Answer
May 2, 2017
Let
# y=arcsinx iff x=siny #
Then using
# sin^2y+cos^2y = 1 => x^2+cos^2y=1 #
# :. cos^2y = 1 -x^2 #
# :. sec^2y = 1/(1 -x^2) #
And, using the trig identity
# tan^2y+1-=sec^2y #
# :. tan^2y = sec^2y - 1#
# " " = 1/(1 -x^2) - 1#
# " " = (1-(1-x^2))/(1 -x^2) #
# " " = (x^2)/(1 -x^2) #
And so:
# :. tany = sqrt((x^2)/(1 -x^2)) #
# " " = x/sqrt(1 -x^2) #
But
# tan(arcsinx) = x/sqrt(1 -x^2) \ \ \ # QED