Question

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

• A. $\tan {80}^{\text{o}}\cot 5^{\text{o}}$
• B. $\tan {80}^{\text{o}}\cot {10}^{\text{o}}$
• C. $\tan {40}^{\text{o}}\cot 5^{\text{o}}$
• D. $\tan {40}^{\text{o}}\cot {10}^{\text{o}}$

Answer 1

The correct option is A $\tan {80}^{\text{o}}\cot 5^{\text{o}}$

$\text{G.E.}=(1+\sec {10}^{\text{o}})(1+\sec {20}^{\text{o}})(1+\sec {40}^{\text{o}})(1+\sec {80}^{\text{o}})=\frac{(1+\cos {10}^{\text{o}})(1+\cos {20}^{\text{o}})(1+\cos {40}^{\text{o}})(1+\cos {80}^{\text{o}})}{\cos {10}^{\text{o}}\cdot \cos {20}^{\text{o}}\cdot \cos {40}^{\text{o}}\cdot \cos {80}^{\text{o}}}=\frac{2^4{\cos }^2{5}^{\text{o}}\cdot {\cos }^2{10}^{\text{o}}\cdot {\cos }^2{20}^{\text{o}}\cdot {\cos }^2{40}^{\text{o}}}{\cos {10}^{\text{o}}\cdot \cos {20}^{\text{0}}\cdot \cos {40}^{\text{o}}\cdot \cos {80}^{\text{o}}}=16\times \frac{{\cos }^2{5}^{\text{o}}}{\cos {80}^{\text{o}}}(\cos {10}^{\text{o}}\cdot \cos {20}^{\text{o}}\cdot \cos {40}^{\text{o}})=16\times \frac{{\cos }^2{5}^{\text{o}}}{\cos {80}^{\text{o}}}\frac{\sin {80}^{\text{o}}}{8\sin {10}^{\text{o}}}(\cos \theta \cdot \cos 2\theta \cdot \cos 2^2\theta \cdots \cos 2^n\theta =\frac{\sin \left(2^{n+1}\theta \right)}{2^{n+1}\sin \theta })=\frac{2\tan {80}^{\text{o}}\cdot {\cos }^2{5}^{\text{o}}}{2\sin 5^{\text{o}}\cdot \cos 5^{\text{o}}}=\tan {80}^{\text{o}}\cot 5^{\text{o}}$