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  4. If x sin θ+y cos θ=2 and x sin θ-y cos θ=0 then (1/x2)+(1/y2) is

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Q. If $x \sin \theta+y \cos \theta=2$ and $x \sin \theta-y \cos \theta=0$ then $\frac{1}{x^2}+\frac{1}{y^2}$ is

Trigonometry

Solution:

Given: $x \sin \theta-y \cos \theta=0$
$ \Rightarrow x \sin \theta=y \cos \theta $
$ \text { and } x \sin \theta+y \cos \theta=2 $
$\Rightarrow y \cos \theta+y \cos \theta=2 \Rightarrow 2 y \cos \theta=2$
$ \Rightarrow y \cos \theta=1 $
$ \therefore x \sin \theta=1 \text { and } y \cos \theta=1$
$ \sin \theta=\frac{1}{x} \cos \theta=\frac{1}{y}$
Squaring the above equations, we get
$ \sin ^2 \theta=\frac{1}{x^2} $...(i)
$ \cos ^2 \theta=\frac{1}{y^2}$...(ii)
Adding equations (i) and (ii)
$\sin ^2 \theta+\cos ^2 \theta=\frac{1}{x^2}+\frac{1}{y^2}=1$