how do you simplify this inverse fraction

Question

how am i supposed to simplify an inverse trig function when it has an x in the numerator? This is the question that i need help on tan(arcsinx/4) can you please help me solve it thank you

Answer 1

Remember that you can write y = tan(arcsin(x/4)) as y = sin(arcsin(x/4)) / cos(arcsin(x/4)) because tan(x) = sin(x)/cos(x). The sin and the arcsin on top will cancel leaving you with y = x/(4*cos(arcsin(x/4)).

To further simplify the denominator cos(arcsin(x/4)) draw a right triangle. arcsin(x/4) = z -> x/4 = sin(z), where z is the (unknown) interior angle of the right triangle in radians. Then x/4 = opposite over hypotenuse of that triangle, so label those. Use Pythagorean theorem to then find and label the adjacent angle on the triangle, too.

To finish up simplifying x/(4*cos(arcsin(x/4)) = x/(4*cos(z)) just look at the right triangle with the three sides now labeled and recall cos(z) = adjacent/hypotenuse. Find that fraction and then plug it back in and you should have a simplified algebraic equation for y in terms of x with no trigonometric functions involved. There will only be a certain x domain where the expression is real-valued, however.