Verify the identity
$$\csc^6 \alpha - \cot^6 \alpha = 1+3\csc^2\alpha \cot^3\alpha $$

Question

Verify the identity
$$\csc^6 \alpha - \cot^6 \alpha = 1+3\csc^2\alpha \cot^3\alpha  $$

Uniteasy Admin · Accepted Answer

$$\csc^6 \alpha - \cot^6 \alpha $$
$$=\dfrac{1}{\sin^6\alpha } -\dfrac{\cos^6\alpha }{\sin^6\alpha } $$
$$=\dfrac{\sin^6-\cos^6\alpha }{\sin^6\alpha } $$
$$ 1+3\csc^2\alpha \cot^3\alpha$$
$$=1+3\dfrac{1}{\sin^2} \dfrac{\cos^3\alpha }{\sin^3\alpha } $$
$$ \sin^6-\cos^6\alpha=\sin^6\alpha +3\sin\alpha \cos^3\alpha  $$
$$\cos^3\alpha +3\sin \alpha =0 $$

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