How do you simplify the expression #sin(2 arctan x) #?
Let's assume by
The double angle formula for sine is
We can think of
The sign of sine and of cosine are each ambiguous when we know the tangent. But the sign of the product sine and cosine and the sign of the tangent (which is the quotient of sine and cosine) are the same.
Despite the ambiguity of the sign of the factors, the sign of the sine of the double angle is not ambiguous.