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Question

If x sin3θ+y cos3θ=sinθ cosθ and x sinθ=y cosθ, then x2+y2= ___.



A

-1

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B

0

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C

1

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D

2

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Solution

The correct option is D

1


Given: x sin3θ+y cos3θ=sinθ cos θ
x sinθ(sin2θ)+y cos3θ=sinθ cosθ.....(i)

We know, x sinθ=y cosθ... (given)
Substitute in (i), we get,
y cos θsin2θ+y cos3θ=sinθ cosθ
y cosθ(sin2θ+cos2θ)=sinθ cosθ
y cosθ=sinθ cosθy=sinθ....(ii)

Again, from (i),
x sin3θ+y cos3θ=sinθ cosθ
x sin3θ+y cosθ(cos2θ)=sinθ cosθ

We know, y cosθ=x sinθ ....(given)
On substituing, we get,
x sin3θ+x sinθ cos2θ=sinθ cosθ
x sinθ(sin2θ+cos2θ)=sinθ cosθ
x sinθ=sinθ cosθ
x=cosθ....(iii)

Squaring and adding eqn (ii) and (iii) we get,

x2+y2=cos2θ+sin2θ

x2+y2=1


Mathematics

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