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Question

$$\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} ... \tan 89^{\circ} = $$


A
1
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B
1
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C
0
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D
None of above
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Solution

The correct option is D $$1$$
$$(\tan 1^{\circ} \tan 89^{\circ}) (\tan 2^{\circ} \tan 88^{\circ}) ..... (\tan 44^{\circ} \tan 46^{\circ}) \tan 45^{\circ}$$, group pair of terms 
$$= \left \{\tan 1^{\circ} \tan (90^{\circ} - 1^{\circ})\right \} \left \{\tan 2^{\circ} \tan (90^{\circ} - 2^{\circ})\right \} ...... \left \{\tan 44^{\circ} \tan (90^{\circ} - 44^{\circ})\right \}\tan 45^{\circ}$$
$$= (\tan 1^{\circ} \cot 1^{\circ})(\tan 2^{\circ} \cot 2^{\circ}) ..... (\tan 44^{\circ} \cot 44^{\circ}) \tan 45^{\circ}$$
$$= (1 \times 1 ..... 1)\times 1 = 1$$
Note that $$\tan\theta\times \cot\theta=1$$

Mathematics

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