simplify the following expression tan(sin^-1(2x))
draw a triangle
opposite the angle θ is side 2x
hypotenuse is 1
sinθ = 2x/1
tanθ = 2x/√(1-4x2)
To simplify the expression tan(sin^-1(2x)), we can use the trigonometric identity:
sin^2(x) + cos^2(x) = 1
First, let's start by finding sin(sin^-1(y)), which will help us simplify our expression. In this case, y is 2x, so we want to find sin(sin^-1(2x)):
sin(sin^-1(2x)) = 2x
Now we substitute this value back into the original expression:
tan(sin^-1(2x)) = tan(sin(sin^-1(2x))) = tan(2x)
Therefore, the simplified expression is tan(2x).