Consider the following: I. 1 − sin⁶α = cos²α(cos⁴α − 3cos²α + 3). II. cos⁸α − sin⁸α = 2sin²α(1 − cos⁴α + sin²α cos²α). Which of the above is/are identities?
A.I only✓ Correct
B.II only
C.Both I and II
D.Neither I nor II
Explanation
Statement I: 1 − sin⁶α = (1 − sin²α)(1 + sin²α + sin⁴α) = cos²α(1 + sin²α + sin⁴α). Substituting sin²α = 1 − cos²α: 1 + (1 − cos²α) + (1 − cos²α)² = 1 + 1 − cos²α + 1 − 2cos²α + cos⁴α = 3 − 3cos²α + cos⁴α = cos⁴α − 3cos²α + 3. So I is correct. Statement II: cos⁸α − sin⁸α = (cos⁴α − sin⁴α)(cos⁴α + sin⁴α) = (cos²α − sin²α)(cos²α + sin²α)(cos⁴α + sin⁴α) = (cos²α − sin²α)(cos⁴α + sin⁴α). This is not equal to the given RHS, so II is incorrect.
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