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Principal Solution of Trigonometric Equation

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Question

Evaluate ${tan}^{2} \frac{π}{16} + {tan}^{2} \frac{2 π}{16} + {tan}^{2} \frac{3 π}{16} + . . . . + {tan}^{2} \frac{7 π}{16}$ .

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Solution

$\frac{7 π}{16} = \frac{π}{2} - \frac{π}{16}$

$\frac{6 π}{16} = \frac{π}{2} - \frac{2 π}{16}$

$\frac{5 π}{16} = \frac{π}{2} - \frac{3 π}{16}$

$\frac{4 π}{16} = \frac{π}{4}$

L.H.S $= {tan}^{2} \frac{π}{16} + {tan}^{2} \frac{2 p i}{16} + {tan}^{2} \frac{3 π}{16} + {tan}^{2} \frac{4 π}{16} + {tan}^{2} \frac{5 π}{16} + {tan}^{2} \frac{6 π}{16} + {tan}^{2} 7 \frac{π}{16}$

$= {tan}^{2} \frac{π}{16} + {tan}^{2} \frac{2 p i}{16} + {tan}^{2} \frac{3 π}{16} + {tan}^{2} \frac{π}{4} + {tan}^{2} (\frac{π}{2} - \frac{3 π}{16}) + {tan}^{2} (\frac{π}{2} - \frac{2 π}{16}) + {tan}^{2} (\frac{π}{2} - \frac{π}{16})$

$= {tan}^{2} \frac{π}{16} + {tan}^{2} \frac{2 p i}{16} + {tan}^{2} \frac{3 π}{16} + {tan}^{2} \frac{π}{4} + {cot}^{2} \frac{3 π}{16} + {cot}^{2} \frac{2 π}{16} + {cot}^{2} \frac{π}{16}$

$= {tan}^{2} \frac{π}{16} + {cot}^{2} \frac{π}{16} + {tan}^{2} \frac{2 π}{16} + {cot}^{2} \frac{2 π}{16} + {tan}^{2} \frac{3 π}{16} + {cot}^{2} \frac{3 π}{16} + 1$

$= {(tan \frac{π}{16} + cot \frac{π}{16})}^{2} - 2 tan \frac{π}{16} cot \frac{π}{16} + {(tan \frac{2 π}{16} + cot \frac{2 π}{16})}^{2} - 2 tan \frac{2 π}{16} cot \frac{2 π}{16} + {(tan \frac{3 π}{16} + cot \frac{3 π}{16})}^{2} - 2 tan \frac{3 π}{16} cot \frac{3 π}{16} + 1$

$= - 5 + {⎛ ⎜ ⎜ ⎝ \frac{sin \frac{π}{16}}{cos \frac{π}{16}} + \frac{cos \frac{π}{16}}{sin \frac{π}{16}} ⎞ ⎟ ⎟ ⎠}^{2} + {⎛ ⎜ ⎜ ⎜ ⎝ \frac{sin \frac{2 π}{16}}{cos \frac{2 π}{16}} + \frac{cos \frac{2 π}{16}}{sin \frac{2 π}{16}} ⎞ ⎟ ⎟ ⎟ ⎠}^{2} + {⎛ ⎜ ⎜ ⎜ ⎝ \frac{sin \frac{3 π}{16}}{cos \frac{3 π}{16}} + \frac{cos \frac{3 π}{16}}{sin \frac{3 π}{16}} ⎞ ⎟ ⎟ ⎟ ⎠}^{2}$

$= - 5 + {⎛ ⎜ ⎜ ⎝ \frac{1}{sin \frac{π}{16} cos \frac{π}{16}} ⎞ ⎟ ⎟ ⎠}^{2} + {⎛ ⎜ ⎜ ⎜ ⎝ \frac{1}{sin \frac{2 π}{16} cos \frac{2 π}{16}} ⎞ ⎟ ⎟ ⎟ ⎠}^{2} + {⎛ ⎜ ⎜ ⎜ ⎝ \frac{1}{sin \frac{3 π}{16} cos \frac{3 π}{16}} ⎞ ⎟ ⎟ ⎟ ⎠}^{2}$

$= - 5 + {⎛ ⎜ ⎜ ⎝ \frac{2}{sin \frac{π}{8}} ⎞ ⎟ ⎟ ⎠}^{2} + {⎛ ⎜ ⎜ ⎜ ⎝ \frac{2}{sin \frac{2 π}{8}} ⎞ ⎟ ⎟ ⎟ ⎠}^{2} + {⎛ ⎜ ⎜ ⎜ ⎝ \frac{2}{sin \frac{3 π}{8}} ⎞ ⎟ ⎟ ⎟ ⎠}^{2}$

$= - 5 + \frac{8}{1 - cos \frac{π}{4}} + \frac{8}{1 - cos \frac{π}{2}} + \frac{8}{1 - cos \frac{3 π}{4}}$

$= - 5 + \frac{8 \sqrt{2}}{\sqrt{2} - 1} + 8 + 8 \sqrt{2} (\sqrt{2} - 1)$

$= - 5 + 8 \sqrt{2} (\sqrt{2} - 1) + 8 \sqrt{2} (\sqrt{2} - 1)$

$= 35$

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Similar questions

Q. Find the exact value of:
(a)

t a n^{2} \frac{π}{16} + t a n^{2} \frac{2 π}{16} + t a n^{2} \frac{3 π}{16} + . . . . . . + t a n^{2} \frac{7 π}{16}

{(\frac{2 + s e c 20^{o}}{c o s e c 20^{o}})}^{2}

t a n^{2} \frac{π}{16} + t a n^{2} \frac{2 π}{16} + . . . . + t a n^{2} \frac{6 π}{16} + t a n^{2} \frac{7 π}{16} = 35

Q. The value of

{tan}^{2} \frac{π}{7} + {tan}^{2} \frac{2 π}{7} + {tan}^{2} \frac{3 π}{7}

List I

List II

and select the correct answer using the code given below the lists :

\begin{matrix} List I & List II (A) & 4 ({sin}^{6} \frac{π}{16} + {sin}^{6} \frac{3 π}{16} + {sin}^{6} \frac{5 π}{16} + {sin}^{6} \frac{7 π}{16}) equals & (P) & 2 (B) & If α = \frac{2 π}{7}, then the value of 5 + sec α + sec 2 α + sec 4 α is & (Q) & 4 (C) & {tan}^{2} \frac{π}{16} + {tan}^{2} \frac{3 π}{16} + {tan}^{2} \frac{5 π}{16} + {tan}^{2} \frac{7 π}{16} - 24 equals & (R) & 5 (D) & The maximum value of & (S) & 1 2 (7 cos θ + 24 sin θ) \times (7 sin θ - 24 cos θ) - 623 for θ \in R is \end{matrix}

Which of the following is the only CORRECT combination?

{tan}^{2} \frac{π}{16} + {tan}^{2} \frac{2 π}{16} + {tan}^{2} \frac{3 π}{16} + \dots \dots + {tan}^{2} \frac{7 π}{16}

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