Question

If $tan\theta +sin\theta =mandtan\theta -sin\theta =n,showthat{m}^2-n^2=4\sqrt{mn}.$

Answer 1

$tan\theta +sin\theta =m$ $tan\theta -sin\theta =n$

$m^2-n^2=4\sqrt{mn}$

$\Rightarrow (tan\theta +sin\theta {)}^2-(tan\theta -sin\theta {)}^2=4\sqrt{tan{\theta }^2-si{n}^2\theta }$

$\Rightarrow 4tan\theta .sin\theta =4\sqrt{sin\theta (\frac{1}{co{s}^2\theta }-1)}$

$=4\sqrt{si{n}^2\theta \left(\frac{1-co{s}^2\theta }{co{s}^2\theta }\right)}$

$=4\sqrt{si{n}^2\theta +ta{n}^2\theta }$

$=4sin\theta tan\theta$ HP