How do I find the inverse of this trigonometry problem?

Question

cot²x/(1 + csc² x) . To get the denominator to 1, what do I multiply by?

Answer 1

I find that converting everything to sines and cosines in a problem like this often leads to a simplification I can recognize. For example:

y(x) = cot²(x) / (1 + csc²(x)) ... becomes:
y(x) = (cos²(x) / sin²(x)) / (1 + 1/sin²(x))
y(x) = cos²(x) / (sin²(x) + 1)

Then use Pythagoras (sin²(x) + cos²(x) = 1) and/or the definition of the tangent (and others, in terms of sine and cosine) to simplify, if possible. Here, converting to cosines gives:

y(x) = cos²(x) / (2 - cos²(x))

and the solution follows easily as a function of either an inverse cosine or inverse secant. Take your pick.