cot(x) = sqrt[cosec^2(x) - 1]
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One plus cosecant squared x is equal to cotangent squared x.
No.
sin(x) = [1 + cot^2(x)]^-0.5
The inverse of sine (sin) is cosecant (csc). The inverse of cosine (cos) is secant (sec). The inverse of tangent (tan) is cotangent (cot).
The answer depends very much on what level of mathematics you are at. At the very basic level, these ratios are defined only for right angled triangles. Given an angle x, let A denote the length of the side adjacent to it, O denote the opposite side and H the hpotenuse. All three lengths must be measured in the same units. Then Cosine x = A/H Sine x = O/H Tan x = O/A Secant x = H/A = 1/Cosine x Cosecant x = H/O = 1/Sine x Cotangent x = A/O = 1/Tan x At a more advanced level, these ratios will be defined as infinte convergent series.