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Question
If x sin3 θ + y cos3 θ = sin θ cos θ and x sin θ = y cos θ, then prove that x2 + y2 = 1
Solution
Given x sin2 θ + y cos2 θ = sin θ cos θ
x sin θ = y cos θ ...(1)
x sin3 θ + y cos3 θ = sin θ cos θ
x sin θ (sin2 θ) + y cos θ (cos2 θ) = sin θ cos θ
x sin θ (sin2 θ) + x sin θ (cos2 θ) = sin θ cos θ
x sin θ (sin2 θ + cos2 θ) = sin θ cos θ
x sin θ = sin θ cos θ
x = cos θ
substitute x = cos θ in (1)
cos θ sin θ = y cos θ y = sin θ
L.H.S = x2 + y2 = cos2 θ + sin2 θ = 1
L.H.S = R.H.S
Hence it is proved.
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