Question

If $A + B = \frac{\pi}{2}$ then the maximum value of $\cos A \cdot \cos B$ is

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{1}{2}$
• C. $-\frac{1}{2}$
• D. $-\frac{1}{\sqrt{2}}$

Hint:

If sum of two variables is constant, their product is maximum when they are equal.

Answer 1

The Correct Option is B

Step 1: Concept
Use the product-to-sum identity: $\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$.

Step 2: Meaning
Since $A+B = \pi/2$, $\cos(A+B) = 0$.

Step 3: Analysis
The expression becomes $\frac{1}{2}[0 + \cos(A-B)] = \frac{1}{2}\cos(A-B)$. The maximum value of the cosine function is 1, which occurs when $A-B = 0$ (i.e., $A = B = \pi/4$).

Step 4: Conclusion
Maximum value $= \frac{1}{2}(1) = \frac{1}{2}$. Final Answer: (B)