$Write the value of \tan 1 ° \tan 2 ° . . . \tan 89 ° .$

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Solution

$\tan 1 ° \tan 2 ° . . . \tan 89 ° = \tan 1 ° \tan 2 ° \tan 3 ° . . . \tan 45 ° . . . \tan 87 ° \tan 88 ° \tan 89 ° = \tan 1 ° \tan 2 ° \tan 3 ° . . . \tan 45 ° . . . \cot (90 ° - 87 °) \cot (90 ° - 88 °) \cot (90 ° - 89 °) = \tan 1 ° \tan 2 ° \tan 3 ° . . . \tan 45 ° . . . \cot 3 ° \cot 2 ° \cot 1 ° = \tan 1 ° \times \tan 2 ° \times \tan 3 ° \times . . . \times 1 \times . . . \times \frac{1}{\tan 3 °} \times \frac{1}{\tan 2 °} \times \frac{1}{\tan 1 °} = 1$

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