Question:
The value of $\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}$ is
(a) −1
(b) 0
(c) 1
(d) None of these
Solution:
Here we have to find: $\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}$
Now
$\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}$
$=\tan \left(90^{\circ}-80^{\circ}\right) \tan \left(90^{\circ}-75^{\circ}\right) \tan 75^{\circ} \tan 80^{\circ}$
$=\cot 80^{\circ} \cot 75^{\circ} \tan 75^{\circ} \tan 80^{\circ}$
$=\left(\cot 80^{\circ} \tan 80^{\circ}\right)\left(\cot 75^{\circ} \tan 75^{\circ}\right)$
$=1 \times 1$
$=1$$[$ Since $\cot \theta \tan \theta=1]$
Hence the correct option is $(c)$