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TS EAMCET · Maths · Trigonometric Ratios & Identities

\(\operatorname{coth}^2 x-\tanh ^2 x=\)

  1. A \(4 \operatorname{cosech} 2 x \tanh 2 x\)
  2. B \(4 \operatorname{sech} 2 x \operatorname{coth} 2 x\)
  3. C \(4 \operatorname{sech} 2 x \tanh 2 x\)
  4. D \(4 \cosh 2 x(\operatorname{cosech} 2 x)^2\)
Verified Solution

Answer & Solution

Correct Answer

(D) \(4 \cosh 2 x(\operatorname{cosech} 2 x)^2\)

Step-by-step Solution

Detailed explanation

\(\operatorname{coth}^2 x - \tanh^2 x = \frac{\cosh^2 x}{\sinh^2 x} - \frac{\sinh^2 x}{\cosh^2 x}\) \(= \frac{\cosh^4 x - \sinh^4 x}{\sinh^2 x \cosh^2 x} = \frac{(\cosh^2 x - \sinh^2 x)(\cosh^2 x + \sinh^2 x)}{(\sinh x \cosh x)^2}\)…