\(\dfrac{sin\alpha+cos\alpha}{sin^3\alpha}=\dfrac{1}{sin^2\alpha}\left(\dfrac{sin\alpha}{sin\alpha}+\dfrac{cos\alpha}{sin\alpha}\right)=\left(1+cot^2\alpha\right)\left(1+cot\alpha\right)\)
\(=\dfrac{\left(1+cot^2\alpha\right)\left(1+cot\alpha\right)\left(1-cot\alpha\right)}{1-cot\alpha}=\dfrac{\left(1+cot^2\alpha\right)\left(1-cot^2\alpha\right)}{1-cot\alpha}\)
\(=\dfrac{1-cot^4\alpha}{1-cot\alpha}\)
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