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The value of tan 20 $°$ tan 80 $°$ cot 50 $°$ is ____.

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Detailed Solution

We know that,
tan (a + b) =

\frac{tan a + tan b}{1 - tan atan b}

and tan (a - b) =

\frac{tan a - tan b}{1 + tan atan b}

Thus, tan 20

°

tan 80

°

cot 50

°

= tan (30 – 10)

°

tan (90 – 10)

°

\frac{1}{\tan 50 °}

= tan (30 – 10)

°

tan (90 – 10)

°

\frac{1}{\tan 50 °}

= tan (30 – 10)

°

. cot 10

°

\frac{1}{\tan (60 - 10) °}

= \frac{\tan 30 ° - \tan 10 °}{1 + \tan 30 ° \tan 10 °}

\frac{1}{\tan \tan 10 °}

\frac{1 + \tan 60 ° \tan 10 °}{\tan 60 ° - \tan 10 °}

= \frac{- \tan 10 °}{1 + \tan 10 °}

\frac{1}{\tan 10 °}

\frac{1 + \tan 10 °}{- \tan 10 °}

= \frac{- \tan 10 °}{+ \tan 10 °}

\frac{1}{\tan 10 °}

\frac{1 + \tan 10 °}{- \tan 10 °}

= \frac{1 - 310 °}{3 - \tan^{2} 10 °}

\frac{1}{\tan \tan 10 °}

= \frac{1 - 310 °}{3 \tan \tan 10 ° - \tan^{3} 10 °}

= \frac{1}{\tan \tan 3 (10 °)}

[tan (3A) =

\frac{3 \tan tan A - 3 A}{1 - 3 A}

]

= \frac{1}{\tan \tan 30 °}

= \cot \cot 30 °

= \sqrt{3}

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The value of tan 20° tan 80° cot 50° is ____.

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The value of tan 20 $°$ tan 80 $°$ cot 50 $°$ is ____.