cot4θ=cotnθ− nC2cotn−2θ+ nC4cotn−4θ−⋯ nC1cotn−1θ− nC3cotn−3θ+ nC5cotn−5θ−⋯⇒cot4θ=cot4θ−6cot2θ+14cot3θ−4cotθ

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$\cot 4 θ =$

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$\frac{4 \cot^{3} θ - 4 \cot θ}{\cot^{4} θ - 6 \cot^{2} θ + 1}$

$\frac{4 \cot θ - 4 \cot^{3} θ}{1 - 6 \cot^{2} θ + \cot^{4} θ}$

$\frac{\cot^{4} θ - 6 \cot^{2} θ + 1}{4 \cot^{3} θ - 4 \cot θ}$

$\frac{1 - 6 \cot^{2} θ + \cot^{4} θ}{4 \cot θ - 4 \cot^{3} θ}$

answer is C.

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Detailed Solution

$\cot 4 θ = \frac{\cot^{n} θ -^{n} C_{2} \cot^{n - 2} θ +^{n} C_{4} \cot^{n - 4} θ - \dots}{^{n} C_{1} \cot^{n - 1} θ -^{n} C_{3} \cot^{n - 3} θ +^{n} C_{5} \cot^{n - 5} θ - \dots}$

$\Rightarrow \cot 4 θ = \frac{\cot^{4} θ - 6 \cot^{2} θ + 1}{4 \cot^{3} θ - 4 \cot θ}$

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