The value of tan 10° tan 15° tan 75° tan 80° is
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 $(c)$

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