Question

$\tan 5^{\circ }\cdot \tan {25}^{\circ }\cdot \tan {45}^{\circ }\cdot \tan {65}^{\circ }\cdot \tan {85}^{\circ }=$ ________.

• A. $1$
• B. $5$
• C. $2$
• D. $0$

Answer 1

The correct option is A $1$

$\tan 5^{\circ }\cdot \tan {25}^{\circ }\cdot \tan {45}^{\circ }\cdot \tan {65}^{\circ }\cdot \tan {85}^{\circ }$

$=\tan 5^{\circ }\cdot \tan {25}^{\circ }\cdot \left(1\right)\cdot \tan (90-25)\cdot \tan (90-5)$

$=\tan 5^{\circ }\cdot \tan {25}^{\circ }\cdot \left(1\right)\cdot \cot {25}^{\circ }\cdot \cot 5(\because \tan (90-\theta )=\cot \theta )$

$=\tan 5^{\circ }\cdot \cot 5^{\circ }\cdot \tan {25}^{\circ }\cdot \cot {25}^{\circ }$

$=(\tan 5^{\circ }\cdot \frac{1}{\tan 5^{\circ }})(\tan {25}^{\circ }\cdot \frac{1}{\tan {25}^{\circ }})$ .........$(\because \cot \theta =\frac{1}{\tan \theta })$

$=1$.