Trigonometric Series

Geometry Level 4

Find The Value Of

( tan ⁡ 2 π 7 + tan ⁡ 2 2 π 7 + tan ⁡ 2 3 π 7 ) ⋅ ( cot ⁡ 2 π 7 + cot ⁡ 2 2 π 7 + cot ⁡ 2 3 π 7 ) (\tan^2\dfrac{\pi}{7} + \tan^2\dfrac{2\pi}{7} +\tan^2\dfrac{3\pi}{7})\cdot(\cot^2\dfrac{\pi}{7} + \cot^2\dfrac{2\pi}{7} +\cot^2\dfrac{3\pi}{7})


The answer is 105.

Vraj Mehta by Vraj Mehta , 30, India

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1 solution

Kumar Krish
Jan 28, 2019

Take theta= nπ/7. Then 4theta=nπ-3theta. Put tan on both sides you will get an equation of 6 degree. Convert it into 3 degree by taking tan^2theta as x. You will get x^3-21x^2+35x-7=0.

Put here 1/x instead of x then you will get a 3 degree equation in cot^2 theta. The equation is 7x^3-35x^2+21x-1=0.

Write sum of roots from both the equation and multiply it you will get the answer as 21×5=105

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