If  x=secθ−cosθ, y=secnθ−cosnθ, then  (x2+4)(dydx)2=

# If  $x=\mathrm{sec}\theta -\mathrm{cos}\theta ,\text{\hspace{0.17em}}y={\mathrm{sec}}^{n}\theta -{\mathrm{cos}}^{n}\theta$, then  $\left({x}^{2}+4\right){\left(\frac{dy}{dx}\right)}^{2}=$

1. A

${n}^{2}\left({y}^{2}-4\right)$

2. B

${n}^{2}\left(4-{y}^{2}\right)$

3. C

${n}^{2}\left({y}^{2}+4\right)$

4. D

${n}^{2}\left(y+4\right)$

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### Solution:

$x=\mathrm{sec}\theta -\mathrm{cos}\theta$

$\frac{dx}{d\theta }=\mathrm{sec}\theta \mathrm{tan}\theta +\mathrm{sin}\theta$

$=\mathrm{sec}\theta \mathrm{tan}\theta +\mathrm{cos}\theta \mathrm{tan}\theta$

$=\mathrm{tan}\theta \left(\mathrm{sec}\theta +\mathrm{tan}\theta \right)$

$y={\mathrm{sec}}^{n}\theta -{\mathrm{cos}}^{n}\theta$

$\frac{dy}{d\theta }=n{\mathrm{sec}}^{n-1}\theta \mathrm{sec}\theta \mathrm{tan}\theta -n{\mathrm{cos}}^{n-1}\theta \left(-\mathrm{sin}\theta \right)$

$=n{\mathrm{sec}}^{n}\theta \mathrm{tan}\theta +n{\mathrm{cos}}^{n-1}\theta \mathrm{sin}\theta$

$=n{\mathrm{sec}}^{n}\theta \mathrm{tan}\theta +n{\mathrm{cos}}^{n}\theta \mathrm{tan}\theta$

$=n\mathrm{tan}\theta \left({\mathrm{sec}}^{n}\theta +{\mathrm{cos}}^{n}\theta \right)$

$\frac{dy}{dx}=\frac{dy}{d\theta }}{dx}{d\theta }}=\frac{n\left({\mathrm{sec}}^{n}\theta +{\mathrm{cos}}^{n}\theta \right)}{\mathrm{sec}\theta +\mathrm{cos}\theta }$

$\frac{dy}{dx}=\frac{n\left({\mathrm{sec}}^{n}\theta +{\mathrm{cos}}^{n}\theta \right)}{\mathrm{sec}\theta +\mathrm{cos}\theta }$

${\left(\frac{dy}{dx}\right)}^{2}=\frac{{n}^{2}{\left({\mathrm{sec}}^{n}\theta +{\mathrm{cos}}^{n}\theta \right)}^{2}}{{\left(\mathrm{sec}\theta +\mathrm{cos}\theta \right)}^{2}}$

${x}^{2}+4=\left({\mathrm{sec}}^{2}\theta +{\mathrm{cos}}^{2}\theta -2\right)+4$

$={\mathrm{sec}}^{2}\theta +{\mathrm{cos}}^{2}\theta +2$

$={\left(\mathrm{sec}\theta +\mathrm{cos}\theta \right)}^{2}$

$\left({x}^{2}+4\right){\left(\frac{dy}{dx}\right)}^{2}=\frac{{n}^{2}{\left({\mathrm{sec}}^{n}\theta +{\mathrm{cos}}^{n}\theta \right)}^{2}}{{\left(\mathrm{sec}\theta +\mathrm{cos}\theta \right)}^{2}}×{\left(\mathrm{sec}\theta +\mathrm{cos}\theta \right)}^{2}$

$={n}^{2}{\left({\mathrm{sec}}^{n}\theta +{\mathrm{cos}}^{n}\theta \right)}^{2}$

$={n}^{2}\left({\mathrm{sec}}^{2n}\theta +{\mathrm{cos}}^{2n}\theta +2\right)$

$={n}^{2}\left({\mathrm{sec}}^{2n}\theta +{\mathrm{cos}}^{2n}\theta +2+2-2\right)$

$={n}^{2}\left({\left({\mathrm{sec}}^{n}\theta -{\mathrm{cos}}^{n}\theta \right)}^{2}+4\right)$

$={n}^{2}\left({y}^{2}+4\right)$

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