Question

$\underset{\theta \to 0}{\lim }\frac{{\cos }^2(1-{\cos }^2\left(1-{\cos }^2\left(1\dots {\cos }^2\theta \right)\right)}{\sin \left(\frac{\pi (\sqrt{\theta +4}-2)}{\theta }\right)}=$

• A. $1$
• B. $0$
• C. $\sqrt{2}$
• D. $-\sqrt{2}$

Answer 1

Answer: (C)

$\underset{\theta \to 0}{\lim }\frac{{\cos }^2(1-{\cos }^2\left(1-{\cos }^2\left(1\dots {\cos }^2\theta \right)\right)}{\sin \left(\frac{\pi (\sqrt{\theta +4}-2}{\theta }\right)}$
$=\underset{\theta \to 0}{\lim }\frac{{\cos }^2({\sin }^2\left({\sin }^2\dots \left({\sin }^2\theta \right)\right)}{\sin \left(\frac{\pi (\sqrt{\theta +4}-2}{\theta }\right)}$
$=\underset{\theta \to 0}{\lim }\frac{{\cos }^2({\sin }^2\left({\sin }^2\dots \left({\sin }^2\theta \right)\right)}{\sin \left(\pi \underset{\theta \to 0}{\lim }\frac{\theta }{\theta (\sqrt{\theta +4}+2)}\right)}$
$=\frac{{\cos }^{2}0}{\sin \frac{\pi }{4}}=\sqrt{2}$