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Trigonometric algebraic expansion

CMAT 2018 · Quantitative Techniques & Data Interpretation · Medium · Trigonometry

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If tan⁡A+cot⁡A=5\tan A + \cot A = \sqrt{5}, what is the value of tan⁡3A+cot⁡3A\tan^3 A + \cot^3 A?

  1. A.

    \sqrt{5}

  2. B.

    3

  3. C.

    2\sqrt{5}

  4. D.

    \frac{2}{\sqrt{5}}

Answer

C

Explanation

Let x=tan⁡Ax = \tan A, then cot⁡A=1x\cot A = \frac{1}{x}. Given x+1x=5x + \frac{1}{x} = \sqrt{5}. Using the identity (x+1x)3=x3+1x3+3(x+1x)(x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x}): (5)3=x3+1x3+35  ⟹  55=tan⁡3A+cot⁡3A+35(\sqrt{5})^3 = x^3 + \frac{1}{x^3} + 3\sqrt{5} \implies 5\sqrt{5} = \tan^3 A + \cot^3 A + 3\sqrt{5}. tan⁡3A+cot⁡3A=25\tan^3 A + \cot^3 A = 2\sqrt{5}.

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