If sin⁶ x + cos⁶ x = 7/8, find sin⁴ x + cos⁴ x:
We know that: sin⁶ x + cos⁶ x = (sin² x + cos² x)(sin⁴ x + cos⁴ x - sin² x cos² x). Since sin² x + cos² x = 1, We get: 1 × (sin⁴ x + cos⁴ x - sin² x cos² x) = 7/8. Now, let sin² x cos² x = z. So, sin⁴ x + cos⁴ x = 1 - 2z. From the equation 1 - 2z –z = 7/8, solving gives: 3z = 1 - 7/8 = 1/8, hence z = 1/24. Thus, sin⁴ x + cos⁴ x = 1 - 2 × 1/24 = 1 - 1/12 = 11/12. Correct option: c) 11/12
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