If sinθ + cosθ = m, then sin6θ + cos6θ equals
Explanation:
Given, sinθ + cosθ = m
Taking square boths sides we get sin2θ + cos2θ + 2 × sinθ × cosθ = m2 ⇒ 1 + 2 × sinθ × cosθ = m2 ⇒ sinθ × cosθ = (m2 - 1)/2 ...(1)
Now we need to find the value of sin6θ + cos6θ = (sin2θ)3 + (cos2θ)3 = (sin2θ + cos2θ)(sin4θ + cos4θ - sin2θ × cos2θ) = 1 × ((sin2θ + + cos2θ)2 - 2 × sin2θ × cos2θ - sin2θ × cos2θ) = ((1)2 - 3 × sin2θ × cos2θ) = (1 - 3 × sin2θ × cos2θ) = (1 - 3 × ((m2 - 1)/2)2) = (1 - 3 × (m2 - 1)2/4)
Hence, option (d).
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