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Question Number 108222 by ZiYangLee last updated on 15/Aug/20

Find the value of

\tan 6 ° \tan 42 ° \tan 66 ° \tan 78 ° without

calculator .

Answered by Dwaipayan Shikari last updated on 15/Aug/20

s i n 6 ° s i n 42 ° s i n 66 ° s i n 78 °

= \frac{1}{4} (c o s 60 ° - c o s 72 °) (c o s 36 ° - c o s 120 °)

= \frac{1}{4} (\frac{1}{2} - \frac{\sqrt{5} - 1}{4}) (\frac{\sqrt{5} + 1}{4} + \frac{1}{2})

= \frac{1}{4} (\frac{3 - \sqrt{5}}{4}) (\frac{\sqrt{5} + 3}{4}) = \frac{1}{16}

c o s 6 ° c o s 42 ° c o s 66 ° c o s 78 °

= \frac{1}{4} (c o s 60 ° + c o s 72 °) (c o s 120 ° + c o s 36 °)

= \frac{1}{4} (\frac{1}{2} + \frac{\sqrt{5} - 1}{4}) (- \frac{1}{2} + \frac{\sqrt{5} + 1}{4})

= \frac{1}{4} (\frac{\sqrt{5} + 1}{4}) (\frac{\sqrt{5} - 1}{4}) = \frac{1}{16}

s o

t a n 6 ° t a n 42 ° t a n 66 ° t a n 78 ° = 1