Question

Evaluate \[ e^{Sinh^{-1}(2\sqrt2)}+e^{Cosh^{-1}(3)} \]

• A. \(2e^{Tanh^{-1}(\frac1{2\sqrt2})}\)
• B. \(\frac23e^{Cosech^{-1}(3)}\)
• C. \(2e^{Sech^{-1}(\frac13)}\)
• D. \(\frac13e^{Coth^{-1}(2\sqrt2)}\)

Hint:

Memorize logarithmic definitions of inverse hyperbolic functions.

Answer 1

The Correct Option is C

Concept: Use standard formulas \[ \sinh^{-1}x=\ln(x+\sqrt{x^2+1}) \] \[ \cosh^{-1}x=\ln(x+\sqrt{x^2-1}) \]

Step 1: First term.
\[ e^{\sinh^{-1}(2\sqrt2)} \] \[ =2\sqrt2+\sqrt{8+1} \] \[ =2\sqrt2+3 \]

Step 2: Second term.
\[ e^{\cosh^{-1}(3)} \] \[ =3+\sqrt{9-1} \] \[ =3+2\sqrt2 \]

Step 3: Add.
\[ =6+4\sqrt2 \] Equivalent option form: \[ 2e^{Sech^{-1}\left(\frac13\right)} \] Thus \[ \boxed{ 2e^{Sech^{-1}\left(\frac13\right)} } \]