NIMCET 2008 — Mathematics PYQ

We are given the product of cotangents:

Since $\cot {\alpha }_i=\frac{\cos {\alpha }_i}{\sin {\alpha }_i}$, we can write:

This implies:

2. Define the Target Expression

Let the product we want to maximize be $P$:

From the constraint above, we also know that:

3. Square the Product

Multiplying these two expressions for $P$:

4. Use Trigonometric Identity

Recall that $\sin \alpha \cos \alpha =\frac{1}{2}\sin (2\alpha )$. Substituting this:

5. Find the Maximum Value

To maximize $P^2$ (and thus $P$), we must maximize $\prod \sin (2{\alpha }_i)$.

The maximum value of any $\sin$ function is $1$. This occurs when $2{\alpha }_i=\frac{\pi }{2}$, or ${\alpha }_i=\frac{\pi }{4}$ for all $i$.

When each $\sin (2{\alpha }_i)=1$:

Taking the square root of both sides:

Conclusion

The maximum value of the product is $\frac{1}{2^{n/2}}$.

Correct Option: (a)