Question : If $\operatorname{tan} 15^{\circ}=2-\sqrt{3}$, then the value of $\operatorname{tan} 15^{\circ} \operatorname{cot} 75^{\circ}+\operatorname{tan} 75^{\circ} \operatorname{cot} 15^{\circ}$ is:
Option 1: 6
Option 2: 10
Option 3: 8
Option 4: 14
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Correct Answer: 14
Solution :
Given: $\tan 15°= 2 - \sqrt{3}$
and $\cot 15° = \frac{1}{\tan15°}=\frac{1}{(2 - \sqrt{3})} =2+\sqrt{3}$ [on rationalizing]
$\tan 15° × \cot 75° + \tan 75° × \cot 15°$
= $\tan 15° × \cot (90° – 15°) + \tan (90° – 15°) × \cot 15°$
= $\tan 15° × \tan 15° + \cot 15° × \cot 15°$
= $\tan^2 15°+\cot^2 15°$
= $(2 - \sqrt{3})^2+(2 + \sqrt{3})^2$
= $7 - 4\sqrt{3} + 7 + 4\sqrt{3}$
= $7 + 7$
= $14$
Hence, the correct answer is 14.
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