Sum of Cosines of Angles in Arithmetic Progression

Q. The value of $(1+\sec {10}^{\text{o}})(1+\sec {20}^{\text{o}})(1+\sec {40}^{\text{o}})(1+\sec {80}^{\text{o}})$ is

1. $\tan {80}^{\text{o}}\cot 5^{\text{o}}$
2. $\tan {80}^{\text{o}}\cot {10}^{\text{o}}$
3. $\tan {40}^{\text{o}}\cot 5^{\text{o}}$
4. $\tan {40}^{\text{o}}\cot {10}^{\text{o}}$

Q.

Find the value of cos$\frac{\pi }{6}$.cos$\frac{\pi }{3}$.cos$\frac{4\pi }{6}$.cos$\frac{8\pi }{6}$.cos$\frac{16\pi }{6}$.cos$\frac{32\pi }{6}$

1. $\frac{\sqrt{3}}{32}$

2. $\frac{\sqrt{3}}{64}$

3. $\frac{\sqrt{3}}{8}$

4. $\frac{\sqrt{3}}{16}$