Reciprocals in Radians

Radians change nothing. Take $\csc\left(\frac{7\pi}{4}\right)$: 2π − π/4, quadrant IV, reference π/4. Sine is negative there, so $\csc\left(\frac{7\pi}{4}\right) = -\csc\left(\frac{\pi}{4}\right)$, and sin(π/4) = √2/2 flips to 2/√2 = √2. The value is −√2.

Cotangent flips tangent: cot(π/6) is 1/(√3/3), which is √3, and cot(π/4) is 1. Where tangent is undefined, at π/2, cotangent is 0, because cos/sin is 0/1 there.