If  fθ=tansin-123+cos2θ, then

# $\mathrm{If} \mathrm{f}\left(\mathrm{\theta }\right)=\mathrm{tan}\left({\mathrm{sin}}^{-1}\sqrt{\frac{2}{3+\mathrm{cos}2\mathrm{\theta }}}\right), \mathrm{then}$

1. A

$\mathrm{f}\left(\frac{\mathrm{\pi }}{4}\right)=1$

2. B

$\mathrm{f}\left(\frac{\mathrm{\pi }}{4}\right)=\sqrt{2}$

3. C

$\frac{\mathrm{d}\left(\mathrm{f}\left(\mathrm{\theta }\right)\right)}{\mathrm{d}\left(\mathrm{cos\theta }\right)} \mathrm{at} \mathrm{\theta }=\frac{\mathrm{\pi }}{4} \mathrm{is} -2$

4. D

$\mathrm{f}\text{'}\left(\frac{\mathrm{\pi }}{4}\right)=\sqrt{2}$

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### Solution:

$\begin{array}{l}\mathrm{f}\left(\mathrm{\theta }\right)=\mathrm{tan}\left({\mathrm{sin}}^{-1}\sqrt{\frac{2}{3+\mathrm{cos}2\mathrm{\theta }}}\right)\\ =\mathrm{tan}\left({\mathrm{tan}}^{-1}\sqrt{\frac{2}{1+\mathrm{cos}2\mathrm{\theta }}}\right)\\ =\sqrt{\frac{2}{1+\mathrm{cos}2\mathrm{\theta }}}=\mathrm{sec\theta }\\ ⇒\mathrm{f}\left(\frac{\mathrm{\pi }}{4}\right)=\sqrt{2}\\ \mathrm{f}\text{'}\left(\mathrm{\theta }\right)=\mathrm{sec\theta tan\theta }\\ ⇒\mathrm{f}\text{'}\left(\frac{\mathrm{\pi }}{4}\right)=\sqrt{2}\\ {\frac{\mathrm{d}\left(\mathrm{f}\left(\mathrm{\theta }\right)\right)}{\mathrm{d}\left(\mathrm{cos\theta }\right)}|}_{\mathrm{\theta }=\frac{\mathrm{\pi }}{4}}={\left(\frac{-1}{{\mathrm{cos}}^{2}\mathrm{\theta }}\right)}_{\mathrm{\theta }=\frac{\mathrm{\pi }}{4}}=-2\end{array}$

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