The value of tan 1° tan 2° tan 3° _ tan 89° is _.

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Solution

tan 1° tan 2° tan 3° ........ tan 89°
Since 89° = 90° – 1
88° = 90° – 2 ........... 46° = 90° – 44°
and tan 89° = tan (90° – 1°) = cot 1°
tan 88° = tan (90° – 2°) = cot 2°
.
.
.
tan 46° = tan (90° – 44°) = cot 44°
tan 45° = 1
∴ tan 1° tan 2° tan 3° .......... tan 45° cot 44° .....cot 3° cot 2° cot 1°
$= \tan 1 ° \tan 2 ° \tan 3 ° . . . . . \frac{1}{\tan 4.1 °} . . . . . \frac{1}{\tan 1 °}$
= 1
Hence, tan 1° tan 2° ....... tan 89° = 1.

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