Q) If x sin3 θ + y cos3 θ = sin θ cos θ and x sin θ = y cos θ, prove that x2 + y2 = 1
Ans: Given that x sin3 θ + y cos3 θ = sin θ cos θ
∴ x sin θ sin2 θ + y cos θ cos2 θ = sin θ cos θ …….. (i)
Step 1: since we are given that x sin θ = y cos θ, let’s substitute the same in the equation (i), we get:
∴ 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 [ since sin2 θ + cos2 θ = 1]
∴ x = cos θ
Step 2: Since, we had x sin θ = y cos θ
by substituting x = cos θ in the above, we get:
x sin θ = y (x)
∴ y = sin θ
Step 3: let’s find the value of x 2 + y 2
We just calculated x = cos θ and y = sin θ
∴ x 2 + y 2 = (cos θ) 2 + (sin θ) 2
= cos2 θ + sin2 θ
= 1
∴ x 2 + y 2 = 1
Hence Proved !
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