Q63 / 160 · TS EAMCET 2024 (09 May Shift 1) · Mathematics

If $\tan A \lt 0$ and $\tan 2 A=-\frac{4}{3}$, then $\cos 6 A=$

  • A
    $\frac{117}{125}$
  • Correct answer
    B
    $-\frac{117}{125}$
  • C
    $\frac{120}{169}$
  • D
    $-\frac{120}{169}$

Solution

$\tan 2 A=-\frac{4}{3} \Rightarrow \frac{2 \tan A}{1-\tan ^2 A}=-\frac{4}{3}$ $\begin{aligned} & 2 \tan ^2 A-3 \tan A-2=0 \\ & \tan A=\frac{3+\sqrt{9+16}}{-4}=\frac{-1}{2}, 2 \\ & \because \tan A \lt 0 \Rightarrow \tan A=\frac{-1}{2} \\ & \cos 2 A=\frac{1-\tan ^2 A}{1+\tan ^2 A}=\frac{1-\frac{1}{4}}{1+\frac{1}{4}}=\frac{3}{5} \\ & \cos 6 A=4 \cos ^3 2 A-3 \cos 2 A=4 \times \frac{27}{125}-3 \times \frac{3}{5}=\frac{-117}{125}\end{aligned}$

Hence, the correct answer is option: 'B'.

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