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Question Number 34647 by JOHNMASANJA last updated on 09/May/18

i f {x s i n}^{3} θ + {y c o s}^{3} θ = s i n θ c o s θ

a n d x s i n θ - y c o s θ = 0

p r o v e t h a t x^{2} + y^{2} = 1

Answered by tanmay.chaudhury50@gmail.com last updated on 09/May/18

x s i n θ = y c o s θ

\frac{x}{c o s θ} = \frac{y}{s i n θ} = k (a s s u m e d)

x = k c o s θ, y = k s i n θ

p u t t i n g i n f i r s t e q u a t i o n

k c o s θ {s i n}^{3} θ + k s i n θ {c o s}^{3} θ = s i n θ c o s θ

k s i n θ c o s θ ({s i n}^{2} θ + {c o s}^{2} θ) = s i n θ c o s θ

s o k = 1

h e n c e x^{2} + y^{2}

= {c o s}^{2} θ + {s i n}^{2} θ

= 1

Answered by ajfour last updated on 09/May/18

x \sin θ = y \cos θ

\Rightarrow x \sin θ (\sin^{2} θ) + y \cos θ (\cos^{2} θ)

= \frac{x \sin θ \cos θ}{x}

o r x \sin^{2} θ + x \cos^{2} θ = \cos θ

\Rightarrow x = \cos θ

s i m i l a r l y

y \sin^{2} θ + y \cos^{2} θ = \sin θ

\Rightarrow y = \sin θ

h e n c e x^{2} + y^{2} = 1 .

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