Question

Find the value of $\tan {10}^{\circ }\tan {15}^{\circ }\tan {75}^{\circ }\tan {80}^{\circ }$

• A. $\frac{1}{16}$
• B. $0$
• C. $1$
• D. None of these

Answer 1

Answer: (C)

$1$

Taking the given

$\tan {10}^{\circ }\tan {15}^{\circ }\tan {75}^{\circ }\tan {80}^{\circ }$

Arrange the ratio whose angle sum is 90

$\tan {10}^{\circ }\tan {80}^{\circ }\tan {15}^{\circ }\tan {75}^{\circ }$

$=\tan {10}^{\circ }\left(\tan \right({90}^{\circ }-{10}^{\circ }\left)\right)\tan {15}^{\circ }\left(\tan \right({90}^{\circ }-{15}^{\circ }\left)\right)$

We know that $\tan (90-\theta )=cot\theta$ so we get

$=\tan {10}^{\circ }\cot {10}^{\circ }\cdot \tan {15}^{\circ }\cot {15}^{\circ }$

$cot\theta =\frac{1}{\tan \theta }$

$=\tan {10}^{\circ }.\frac{1}{\tan {10}^{\circ }}.\tan {15}^{\circ }.\frac{1}{\tan {15}^{\circ }}$

$=1\times 1$

$=1$