The value of the expression tan- 7 -2tan2- 7-4tan4- 7-88- 7 is equal to

We know that, θ tanθ = cos ^2 θ sin^2 θsinθcosθ ⇒θ tanθ = 2cos2θsin 2θ = 2 2θ ⇒tanθ = θ 2 2θ⋯(1) Assuming π 7 = θ tanπ 7 +2tan2π 7+4tan4π 7+88π 7 nbsp;= ...

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Standard XIII

Mathematics

Trigonometric Ratios of Allied Angles

The value of ...

Question

The value of the expression $tan \frac{π}{7} + 2 tan \frac{2 π}{7} + 4 tan \frac{4 π}{7} + 8 cot \frac{8 π}{7}$ is equal to

cosec \frac{2 π}{7} + cot \frac{2 π}{7}

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tan \frac{π}{14} - cot \frac{π}{14}

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\frac{sin \frac{2 π}{7}}{1 - cos \frac{2 π}{7}}

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\frac{1 + cos \frac{π}{7} + cos \frac{2 π}{7}}{sin \frac{π}{7} + sin \frac{2 π}{7}}

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Solution

The correct option is D $\frac{1 + cos \frac{π}{7} + cos \frac{2 π}{7}}{sin \frac{π}{7} + sin \frac{2 π}{7}}$
We know that,
$cot θ - tan θ = \frac{{cos}^{2} θ - {sin}^{2} θ}{sin θ cos θ} \Rightarrow cot θ - tan θ = \frac{2 cos 2 θ}{sin 2 θ} = 2 cot 2 θ \Rightarrow tan θ = cot θ - 2 cot 2 θ \dots (1)$
Assuming $\frac{π}{7} = θ$
$tan \frac{π}{7} + 2 tan \frac{2 π}{7} + 4 tan \frac{4 π}{7} + 8 cot \frac{8 π}{7} = tan θ + 2 tan 2 θ + 4 tan 4 θ + 8 cot 8 θ$
Now,
$tan θ = cot θ - 2 cot 2 θ 2 tan 2 θ = 2 (cot 2 θ - 2 cot 4 θ) 4 tan 4 θ = 4 (cot 4 θ - 2 cot 8 θ) 8 cot 8 θ = 8 cot 8 θ \Rightarrow tan θ + 2 tan 2 θ + 4 tan 4 θ + 8 cot 8 θ = cot θ ∴ tan \frac{π}{7} + 2 tan \frac{2 π}{7} + 4 tan \frac{4 π}{7} + 8 cot \frac{8 π}{7} = cot \frac{π}{7}$

Now,
$cosec \frac{2 π}{7} + cot \frac{2 π}{7} = \frac{1 + cos \frac{2 π}{7}}{sin \frac{2 π}{7}} = \frac{2 {cos}^{2} \frac{π}{7}}{2 cos \frac{π}{7} sin \frac{π}{7}} = cot \frac{π}{7}$

$tan \frac{π}{14} - cot \frac{π}{14}$
Using equation $(1)$ ,
$= - 2 cot \frac{2 π}{14} = - 2 cot \frac{π}{7}$

$\frac{sin \frac{2 π}{7}}{1 - cos \frac{2 π}{7}} = \frac{2 sin \frac{π}{7} cos \frac{π}{7}}{2 {sin}^{2} \frac{π}{7}} = cot \frac{π}{7}$

$\frac{1 + cos \frac{π}{7} + cos \frac{2 π}{7}}{sin \frac{π}{7} + sin \frac{2 π}{7}} = \frac{cos \frac{π}{7} + 2 {cos}^{2} \frac{π}{7}}{sin \frac{π}{7} + 2 sin \frac{π}{7} cos \frac{π}{7}} = \frac{cos \frac{π}{7} (1 + 2 cos \frac{π}{7})}{sin \frac{π}{7} (1 + 2 cos \frac{π}{7})} = cot \frac{π}{7}$

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Similar questions

Q. The value of the expression

tan \frac{π}{7} + 2 tan \frac{2 π}{7} + 4 tan \frac{4 π}{7} + 8 cot \frac{8 π}{7}

is equal to

Q. The value of

tan \frac{π}{7} + tan \frac{2 π}{7} + tan \frac{3 π}{7} + tan \frac{4 π}{7} + tan \frac{5 π}{7} + tan \frac{6 π}{7} + tan \frac{7 π}{7} =

Q. the value of

s i n \frac{π}{7} + s i n \frac{2 π}{7} + s i n \frac{3 π}{7}

is equal to :-

Q. The value of

cos \frac{π}{7} + cos \frac{2 π}{7} + cos \frac{3 π}{7} + cos \frac{4 π}{7} + cos \frac{5 π}{7} + cos \frac{6 π}{7} + cos \frac{7 π}{7}

Q. The equation whose roots are

t a n^{2} \frac{2 π}{7}, t a n^{2} \frac{4 π}{7}, t a n^{2} \frac{8 π}{7} (= t a n^{2} \frac{6 π}{7})

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Trigonometric Ratios of Allied Angles

Standard XIII Mathematics

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The value of the expression tanπ7+2tan2π7+4tan4π7+8cot8π7 is equal to

The value of the expression $tan \frac{π}{7} + 2 tan \frac{2 π}{7} + 4 tan \frac{4 π}{7} + 8 cot \frac{8 π}{7}$ is equal to