Question

Consider the following statements
Statement-I: $\cosh^{-1}x = \tanh^{-1}x$ has no solution
Statement-II: $\cosh^{-1}x = \coth^{-1}x$ has only one solution
The correct answer is

• A. Both statements I and II are true
• B. Both statements I and II are false
• C. Statement I is true, but statement II is false
• D. Statement I is false, but statement II is true

Hint:

When asked to find the number of solutions to an equation involving special functions, the first step should always be to determine the domains of the functions involved. The intersection of the domains is the only region where solutions can exist.

Answer 1

The Correct Option is A

Step 1: Analyze Statement-I.
\[ \text{Domain of } \cosh^{-1}x = [1, \infty), \text{Domain of } \tanh^{-1}x = (-1,1) \] No intersection $\implies$ Statement-I is true.

Step 2: Analyze Statement-II.
\[ \text{Domain of } \coth^{-1}x = (-\infty,-1)\cup(1,\infty), \text{Intersection with } [1,\infty) = (1,\infty) \]
Step 3: Solve $\cosh^{-1x = \coth^{-1}x$.}
\[ x = \cosh y = \coth y = \frac{\cosh y}{\sinh y} \implies \sinh y = 1 \implies x = \cosh y = \sqrt{2} \]
Step 4: Conclusion.
Exactly one solution exists $\implies$ Statement-II is true.
Hence, both statements are true.\