Question

The value of \( \cosh \left( \sin^{-1} \left( \sqrt{8} \right) + \cosh^{-1} 5 \right) \) is:

• A. \( \sqrt{6} + 4\sqrt{2} \)
• B. \( 15 + 8\sqrt{3} \)
• C. \( 6\sqrt{6} + 10\sqrt{2} \)
• D. \( 8 - 15\sqrt{3} \)

Hint:

For expressions involving inverse trigonometric and hyperbolic functions, use appropriate identities to simplify and calculate the value.

Answer 1

The Correct Option is B

Step 1: Simplify \(\sinh^{-1}(\sqrt{8})\)
Let \( \theta = \sinh^{-1}(\sqrt{8}) \). Then:
\( \sinh(\theta) = \sqrt{8}. \)
Using the identity \( \cosh^2(\theta) - \sinh^2(\theta) = 1 \), we get:
\( \cosh(\theta) = \sqrt{1 + \sinh^2(\theta)} = \sqrt{1 + 8} = 3. \)
Step 2: Simplify \(\cosh^{-1}(5)\)
Let \( \phi = \cosh^{-1}(5) \). Then:
\( \cosh(\phi) = 5. \)
Using the identity \( \cosh^2(\phi) - \sinh^2(\phi) = 1 \), we get:
\( \sinh(\phi) = \sqrt{\cosh^2(\phi) - 1} = \sqrt{25 - 1} = \sqrt{24} = 2\sqrt{6}. \)
Step 3: Use the Addition Formula for Hyperbolic Cosine
The addition formula for hyperbolic cosine is:
\( \cosh(A + B) = \cosh(A)\cosh(B) + \sinh(A)\sinh(B). \)
Substitute \( A = \theta \) and \( B = \phi \): \( \cosh(\theta + \phi) = \cosh(\theta)\cosh(\phi) + \sinh(\theta)\sinh(\phi). \)
Substitute the known values:
\( \cosh(\theta + \phi) = (3)(5) + (\sqrt{8})(2\sqrt{6}) = 15 + 2\sqrt{48} = 15 + 2 \cdot 4\sqrt{3} = 15 + 8\sqrt{3}. \)
Step 4: Verify the Answer The result \( 15 + 8\sqrt{3} \) corresponds to option 2.