Q64 / 160 · AP EAMCET 2024 (22 May Shift 1) · Mathematics

The value of $x$ such that $\sin \left(2 \tan ^{-1} \frac{3}{4}\right)=\cos \left(2 \tan ^{-1} x\right)$ is

  • A
    7
  • B
    $\frac{3}{7}$
  • Correct answer
    C
    $\frac{1}{7}$
  • D
    $\frac{4}{7}$

Solution

$\begin{aligned} & \sin \left(2 \tan ^{-1} \frac{3}{4}\right)=\cos \left(2 \tan ^{-1} x\right) \\ & \Rightarrow \frac{2 \times \frac{3}{4}}{1+\left(\frac{3}{4}\right)^2}=\frac{1-x^2}{1+x^2}=\frac{24}{25} \\ & \Rightarrow 25-25 x^2=24+24 x^2 \Rightarrow 49 x^2=1 \Rightarrow x=\frac{1}{7}\end{aligned}$

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

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