Question

Eliminate $\theta$, if

$x=3cosec\theta +4cot\theta$,

$y=4cosec\theta -3cot\theta$.

Answer 1

Given that $x=3cosec\theta +4cot\theta$ and $y=4cosec\theta -3cot\theta$

Multiply equation $1$ by $3$ and equation $2$ by $4$ and add both

We get $cosec\theta =\frac{3x+4y}{25}$

Now multiply equation $1$ by $4$ and equation $2$ by $3$ and subtract both

We get $cot\theta =\frac{4x-3y}{25}$

we know that $cose{c}^2\theta -co{t}^2\theta =1$

$\Rightarrow (3x+4y{)}^2-(4x-3y{)}^2=625$