Question:
\( \sinh^{-1} 2 + \sinh^{-1} 3 = x \Rightarrow \cosh x \) is equal to
\( \sinh^{-1} 2 + \sinh^{-1} 3 = x \Rightarrow \cosh x \) is equal to
Show Hint
$\cosh^2 x - \sinh^2 x = 1$.
Updated On: Apr 10, 2026
- $\frac{1}{2}(3\sqrt{5}+2\sqrt{10})$
- $\frac{1}{2}(3\sqrt{5}-2\sqrt{10})$
- $\frac{1}{2}(12+2\sqrt{50})$
- $\frac{1}{2}(12-2\sqrt{50})$
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The Correct Option is C
Solution and Explanation
Step 1: Hyperbolic Identity
$\cosh(A+B) = \cosh A \cosh B + \sinh A \sinh B$.
Step 2: Find values
$\sinh A = 2 \Rightarrow \cosh A = \sqrt{1+2^2} = \sqrt{5}$. $\sinh B = 3 \Rightarrow \cosh B = \sqrt{1+3^2} = \sqrt{10}$.
Step 3: Substitution
$\cosh x = \sqrt{5}\sqrt{10} + (2 \times 3) = \sqrt{50} + 6$.
Step 4: Final Answer
Multiply and divide by 2: $\frac{12 + 2\sqrt{50}}{2}$.
Final Answer: (C)
$\cosh(A+B) = \cosh A \cosh B + \sinh A \sinh B$.
Step 2: Find values
$\sinh A = 2 \Rightarrow \cosh A = \sqrt{1+2^2} = \sqrt{5}$. $\sinh B = 3 \Rightarrow \cosh B = \sqrt{1+3^2} = \sqrt{10}$.
Step 3: Substitution
$\cosh x = \sqrt{5}\sqrt{10} + (2 \times 3) = \sqrt{50} + 6$.
Step 4: Final Answer
Multiply and divide by 2: $\frac{12 + 2\sqrt{50}}{2}$.
Final Answer: (C)
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