If sinθ + cosecθ = \(\sqrt{5}\), then find the value of sin3θ + cosec3θ.
\(2\sqrt{5}\)
Since cosecθ = \(\frac{1}{\sin\theta}\), sinθ × cosecθ = 1.
Using a3 + b3 = (a + b)3 − 3ab(a + b) with a = sinθ, b = cosecθ:
sin3θ + cosec3θ = \((\sqrt{5})^{3} - 3 \times 1 \times \sqrt{5} = 5\sqrt{5} - 3\sqrt{5} = 2\sqrt{5}\).
Hence, the answer is \(2\sqrt{5}\).
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