Question

If $cosec\theta -\sin \theta =m,\sec \theta -\cos \theta =n$, eliminate $\theta$.

Answer 1

$\csc \theta -\sin \theta =m$

$\frac{1}{\sin \theta }-\sin \theta =m$

$\frac{1-{\sin }^2\theta }{\sin \theta }=m$

$\frac{{\cos }^2\theta }{\sin \theta }=m$

$m^2=\frac{{\cos }^4\theta }{{\sin }^2\theta }$

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$\sec \theta -\cos \theta =n$

$\frac{1}{\cos \theta }-\cos \theta =n$

$\frac{1-{\cos }^2\theta }{\cos \theta }=n$

$\frac{{\sin }^2\theta }{\cos \theta }=n$

$n^2=\frac{{\sin }^4\theta }{{\cos }^2\theta }$

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$n(m^2-1)$

$=\frac{{\sin }^2\theta }{\cos \theta }(\frac{{\cos }^4\theta }{{\sin }^2\theta }-1)$

$=\frac{{\sin }^2\theta }{\cos \theta }(\frac{{\cos }^4\theta -{\sin }^2\theta }{{\sin }^2\theta }-1)$

$={\cos }^3\theta -\frac{{\sin }^2\theta }{\cos \theta }$

$=m^{2}n-n$

$\therefore n(m^2-1)=m^{2}n-n$