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AP EAMCET · Maths · Trigonometric Equations

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

  1. A 7
  2. B \(\frac{3}{7}\)
  3. C \(\frac{1}{7}\)
  4. D \(\frac{4}{7}\)
Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{1}{7}\)

Step-by-step Solution

Detailed explanation

\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…