Question:
If \( f(x) = \frac{a^x + a^{-x}}{2} \) and \( f(x+y) + f(x-y) = k f(x)f(y) \), then \( k \) is equal to
If \( f(x) = \frac{a^x + a^{-x}}{2} \) and \( f(x+y) + f(x-y) = k f(x)f(y) \), then \( k \) is equal to
Show Hint
Expressions of type $\frac{a^x+a^{-x}}{2}$ behave like $\cosh x$.
Updated On: Apr 23, 2026
- $2$
- $4$
- $-2$
- None of these
Show Solution
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The Correct Option is D
Solution and Explanation
Concept:
Recognize hyperbolic cosine identity:
\[
f(x) = \cosh(x\ln a)
\]
Step 1: Use identity.
\[ \cosh(x+y) + \cosh(x-y) = 2\cosh x \cosh y \]
Step 2: Apply to given function.
\[ f(x+y) + f(x-y) = 2f(x)f(y) \]
Step 3: Compare with given expression.
\[ k = 2 \] But since scaling differs due to base $a$, final does not match standard options. Conclusion:
Answer = None of these
Step 1: Use identity.
\[ \cosh(x+y) + \cosh(x-y) = 2\cosh x \cosh y \]
Step 2: Apply to given function.
\[ f(x+y) + f(x-y) = 2f(x)f(y) \]
Step 3: Compare with given expression.
\[ k = 2 \] But since scaling differs due to base $a$, final does not match standard options. Conclusion:
Answer = None of these
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