2025 · Q22
Consider the following: (I): 1 - sin⁶α = cos²α ( cos⁴α − 3 cos²α + 3) (II): cos⁸α−sin⁸α = 2sin²α(1 − cos⁴α + sin²α cos²α) Which of the above is/are identities?
Elementary MathematicsTrigonometry
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Explanation
For (I): 1 − sin⁶α = (1 − sin²α)(1 + sin²α + sin⁴α) = cos²α(1 + sin²α + sin⁴α), and substituting sin²α = 1 − cos²α shows 1 + sin²α + sin⁴α = cos⁴α − 3cos²α + 3, so (I) is an identity. For (II), testing α = π/4 gives LHS = 0 but RHS = 1, so (II) is not an identity. Hence only (I) holds, option (a).