Question

If \( \sinh^{-1}(-\sqrt{3}) + \cosh^{-1}(2) = K \), then \( \cosh K = \):

• A. \( \log(2 - \sqrt{3}) \)
• B. \( \log(2 + \sqrt{3}) \)
• C. \( 0 \)
• D. \( 1 \)

Hint:

Always verify that input values lie within the domain of inverse trigonometric functions. If a given value is outside the valid range, consider interpreting the problem within the framework of hyperbolic functions.

Answer 1

The Correct Option is D

We are given: \[ \sin h^{-1}(-\sqrt{3}) + \cos h^{-1}(2) = K \] We need to determine \( \cosh K \). 

--- Step 1: Express \( \sinh^{-1}(-\sqrt{3}) \) in Terms of Logarithm Using the identity: \[ \sinh^{-1} x = \ln \left( x + \sqrt{x^2 + 1} \right) \] \[ \sinh^{-1}(-\sqrt{3}) = \ln \left( -\sqrt{3} + \sqrt{(\sqrt{3})^2 + 1} \right) \] \[ = \ln \left( -\sqrt{3} + \sqrt{4} \right) \] \[ = \ln \left( -\sqrt{3} + 2 \right) \] Thus, \[ \sinh(\sinh^{-1}(-\sqrt{3})) = -\sqrt{3} \]

 Step 2: Compute \( \cosh^{-1}(2) \) Using the identity: \[ \cosh^{-1} x = \ln \left( x + \sqrt{x^2 - 1} \right) \] \[ \cosh^{-1}(2) = \ln \left( 2 + \sqrt{4 - 1} \right) \] \[ = \ln \left( 2 + \sqrt{3} \right) \] Thus, \[ \cosh(\cosh^{-1}(2)) = 2 \]

 Step 3: Compute \( K \) \[ K = \sinh^{-1}(-\sqrt{3}) + \cosh^{-1}(2) \] \[ K = \ln (2 - \sqrt{3}) + \ln (2 + \sqrt{3}) \] Using the logarithm property: \[ \ln a + \ln b = \ln (a \cdot b) \] \[ K = \ln \left( (2 - \sqrt{3})(2 + \sqrt{3}) \right) \] \[ = \ln \left( 4 - 3 \right) = \ln (1) = 0 \] 

Step 4: Compute \( \cosh K \) \[ \cosh K = \cosh(0) = 1 \] 

--- Final Answer: \(\boxed{1}\)