Fundamentally you can get there using the identity
... sin(a)² + cos(a)² = 1
... 1/cosec(a)² + 1/k² = 1 . . . . . substituting for sec(a) and sin(a)
... 1/cosec(a)² = 1 - 1/k² = (k² -1)/k² . . . . subtract 1/k²
... cosec(a)² = k²/(k² -1) . . . . . Multiply by cosec(a)²k²/(k²-1)
... cosec(a) = k/√(k²-1) . . . . . .take the square root
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You can get there slightly faster using the identities tan²=sec²-1; csc=sec/tan.
