Question

If x = a sin $\theta$ and y = b tan $\theta$, then prove that $\frac{a^2}{x^2}-\frac{b^2}{y^2}=1$

Answer 1

We have ,
$L.H.S=\frac{a^2}{x^2}-\frac{b^2}{y^2}$

$\Rightarrow L.H.S=\frac{a^2}{a^{2}si{n}^2\theta }-\frac{b^2}{b^{2}ta{n}^2\theta }$ $[\because x=asin\theta ,y=btan\theta ]$

$\Rightarrow L.H.S=\frac{1}{si{n}^2\theta }-\frac{1}{ta{n}^2\theta }$

$\Rightarrow L.H.S=cose{c}^2\theta -co{t}^2\theta$ $[\because 1+co{t}^2\theta =cose{c}^2\theta \therefore cose{c}^2\theta -co{t}^2\theta =1]$

$\Rightarrow$ LHS $=1=$ RHS

Hence, proved