Question

tan 10° tan 15° tan 75° tan 80° = ?
(a) $\sqrt{3}$
(b) 1
(c) $\frac{1}{\sqrt{3}}$
(d) –1

Answer 1

$$\begin{gathered} \tan 10°\tan 15°\tan 75°\tan 80° \\ =\tan \left(90°-80°\right)\tan 15°\tan \left(90°-15°\right)\tan 80° \\ =\cot 80°\tan 15°\cot 15°\tan 80°\left(\because \tan \left(90°-\theta \right)=\cot \theta \right) \\ =\left(\frac{1}{\tan 80°}\tan 80°\right)\left(\tan 15°\frac{1}{\tan 15°}\right)\left(\because \cot \theta =\frac{1}{\tan \theta }\right) \\ =1 \\ \mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{option}\mathrm{is}\left(\mathrm{b}\right). \end{gathered}$$