Question

State True or False: $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2(1+\cos x)}}}}=2\cos \frac{x}{16}$

• A. True
• B. False

Answer 1

The correct option is A True

Now,

$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2(1+\cos x)}}}}$

$=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2\times 2{\cos }^2\frac{x}{2}}}}}$

$=\sqrt{2+\sqrt{2+\sqrt{2+2\cos \frac{x}{2}}}}$

$=\sqrt{2+\sqrt{2+\sqrt{4{\cos }^2\frac{x}{4}}}}$

$=\sqrt{2+\sqrt{2+2\cos \frac{x}{4}}}$

$=\sqrt{2+\sqrt{4{\cos }^2\frac{x}{8}}}$

$=\sqrt{2+2\cos \frac{x}{8}}$

$=\sqrt{4{\cos }^2\frac{x}{16}}$

$=2\cos \frac{x}{16}$.