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Show that cosec⁶(alpha) - cot⁶(alpha) = 3*cosec²(alpha)*cot²(alpha) + 1. As an identity, which equivalent simplified form does the left side reduce to?

1. 3*cosec²(alpha)*cot²(alpha) + 1
2. 3*cosec²(alpha)*cot²(alpha) - 1
3. cosec²(alpha)*cot²(alpha) + 1
4. 3*cosec²(alpha)*cot²(alpha) + 3

Correct answer: 3*cosec²(alpha)*cot²(alpha) + 1

Solution

Treating cosec² and cot² as a and b, the difference of cubes factors using a - b = 1, leaving 3ab + 1.

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