Question

Let $x=2^{\left(\log _2 3\right)\left(\log _3 4\right) \ldots \ldots \ldots}$, where the last term in exponent is $\log _{99} 100$, then the value of $\cos \left(\frac{x \pi}{4}\right)+\sin \left(\frac{x \pi}{4}\right)$ is equal to

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

Answer 1

Answer: (D)

$ 2^{\log _2 3 \times \log _3 4 \times \ldots \times \log _{99} 100}=2^{\log _2 100}=100 $
$\therefore \cos \frac{x \pi}{4}+\sin \frac{x \pi}{4}=\cos 25 \pi+\sin 25 \pi=-1 $