`y=1/((1+tan theta)^(sin theta-cos theta)+(cot theta)^(cos theta-cot theta))+ 1/((1+tan theta)^(cos theta-sin theta)+(cot theta)^(sin theta-cottheta)) +1/((1+tan theta)^(cos theta-cot theta)+(cot theta)^(cot theta-sin theta))` then `(dy)/(dx)` at `theta=pi/3` is

Options

A. 0
B. 1
C. `sqrt3`
D. None of these

Answer

To solve the problem, we need to find \(\frac{dy}{dx}\) for the given function \(y\) at \(\theta = \frac{\pi}{3}\).

1. **Understanding the Function**:
The function is given as:
\[ y = \frac{1}{(1 + \tan \theta)^{\sin \theta - \cos \theta} + (\cot \theta)^{\cos \theta - \cot \theta}} + \frac{1}{(1 + \tan \theta)^{\cos \theta - \sin \theta} + (\cot \theta)^{\sin \theta - \cot \theta}} + \frac{1}{(1 + \tan \theta)^{\cos \theta - \cot \theta} + (\cot \theta)^{\cot \theta - \sin \theta}} \]
Here, \(y\) is expressed in terms of \(\theta\) only, and there are no \(x\) terms present in the expression.

2. **Identifying the Variables**:
Since \(y\) is a function of \(\theta\) and does not depend on \(x\), we can conclude that \(y\) is constant with respect to \(x\).

3. **Differentiating with Respect to \(x\)**:
When we differentiate a constant with respect to any variable, the result is always zero:
\[ \frac{dy}{dx} = 0 \]

4. **Conclusion**:
Therefore, the value of \(\frac{dy}{dx}\) at \(\theta = \frac{\pi}{3}\) is:
\[ \frac{dy}{dx} = 0 \]