If cosecθ = y/x, what is tan(3π/2 − θ)?

Correct answer: c √y² −x²/x

cosecθ = y/x ⇒ sinθ = x/y; cosθ = √(y² − x²)/y; tan(3π/2 − θ) = cotθ = cosθ/sinθ = √(y² − x²)/x.

Why the other options are wrong

• a √x² −y²/y: x² − y² is negative (cosecθ = y/x means y ≥ x), so this root is not real.
• b √x² −y²/x: x² − y² is negative and the root is not real; it should be y² − x² under the root.
• d √y² −x²/y: √(y² − x²)/y is cosθ; tan(3π/2 − θ) = cotθ = cosθ/sinθ, whose denominator is x.