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  4. displaystyle lim θ arrow 0 ( cos 2(1- cos 2(1- cos 2(1 ldots cos 2 θ))./ sin ((π(√θ+4-2))θ))=

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Q. $\displaystyle\lim _{\theta \rightarrow 0} \frac{\cos ^{2}\left(1-\cos ^{2}\left(1-\cos ^{2}\left(1 \ldots \cos ^{2} \theta\right)\right)\right.}{\sin \left(\frac{\pi(\sqrt{\theta+4}-2)}{\theta}\right)}=$

Limits and Derivatives

Solution:

$\displaystyle\lim _{\theta \rightarrow 0} \frac{\cos ^{2}\left(1-\cos ^{2}\left(1-\cos ^{2}\left(1 \ldots \cos ^{2} \theta\right)\right)\right.}{\sin \left(\frac{\pi(\sqrt{\theta+4}-2}{\theta}\right)}$
$=\displaystyle\lim _{\theta \rightarrow 0} \frac{\cos ^{2}\left(\sin ^{2}\left(\sin ^{2} \ldots\left(\sin ^{2} \theta\right)\right)\right.}{\sin \left(\frac{\pi(\sqrt{\theta+4}-2}{\theta}\right)}$
$=\displaystyle\lim _{\theta \rightarrow 0} \frac{\cos ^{2}\left(\sin ^{2}\left(\sin ^{2} \ldots\left(\sin ^{2} \theta\right)\right)\right.}{\sin \left(\pi \displaystyle\lim _{\theta \rightarrow 0} \frac{\theta}{\theta(\sqrt{\theta+4}+2)}\right)}$
$=\frac{\cos ^{2} 0}{\sin \frac{\pi}{4}}=\sqrt{2}$