Question

$\sec \theta =\frac{\sqrt{p^2+q^2}}{q}$
then find the value of $\frac{p\sin \theta -q\cos \theta }{p\sin \theta +q\cos \theta }$

Answer 1

$\sec \theta =\frac{\sqrt{p^2+q^2}}{q}$

$\sin \theta =\frac{p}{\sqrt{p^2+q^2}}$

$⟹p\sin \theta =\frac{p^2}{\sqrt{p^2+q^2}}$

$\cos \theta =\frac{q}{\sqrt{p^2+q^2}}$

$⟹q\cos \theta =\frac{q^2}{\sqrt{p^2+q^2}}$

$\frac{p\sin \theta -q\cos \theta }{p\sin \theta +q\cos \theta }=\frac{\frac{p^2}{\sqrt{p^2+q^2}}-\frac{q^2}{\sqrt{p^2+q^2}}}{\frac{p^2}{\sqrt{p^2+q^2}}+\frac{q^2}{p^2+q^2}}=\frac{p^2-q^2}{p^2+q^2}$