Question:

Step 1: To find the correct combination, we first need to analyze the given trigonometric equation and simplify it to a form that relates to the values given in Column-2 and Column-3. The equation provided is $\log 2 - 2\log\sec\theta = 2\log\cosec\theta - \log 2$. We will start by simplifying this equation using logarithmic properties. Step 2: We apply the properties of logarithms to simplify the given equation. Using the property that $\log a - \log b = \log \frac{a}{b}$ and $a\log b = \log b^a$, we can rewrite the equation as $\log \frac{2}{\sec^2\theta} = \log \frac{\cosec^2\theta}{2}$. This simplification allows us to equate the expressions inside the logarithms. Step 3: By equating the expressions inside the logarithms, we get $\frac{2}{\sec^2\theta} = \frac{\cosec^2\theta}{2}$. Knowing that $\sec\theta = \frac{1}{\cos\theta}$ and $\cosec\theta = \frac{1}{\sin\theta}$, we can rewrite this equation in terms of $\sin\theta$ and $\cos\theta$ to further simplify and solve for $\theta$. Step 4: Substituting the trigonometric identities for $\sec\theta$ and $\cosec\theta$, the equation becomes $\frac{2}{1/\cos^2\theta} = \frac{1/\sin^2\theta}{2}$. Simplifying this yields $2\cos^2\theta = \frac{1}{2\sin^2\theta}$, which can be further rearranged to $4\cos^2\theta\sin^2\theta = 1$. Recognizing that $\sin 2\theta = 2\sin\theta\cos\theta$, we can relate this to the double angle formula. Step 5: The equation $4\cos^2\theta\sin^2\theta = 1$ can be written as $(2\sin\theta\cos\theta)^2 = 1$, which simplifies to $\sin^2 2\theta = 1$. This implies that $\sin 2\theta = \pm 1$. However, for $\sin 2\theta = 1$, the solutions are $\theta = 2n\pi + \frac{\pi}{4}$, where $n$ is an integer. Step 6: Given $\theta = 2n\pi + \frac{\pi}{4}$, we can now find the value of $\sin\theta + \cos\theta$. Substituting $\theta = \frac{\pi}{4}$ (for $n=0$) into $\sin\theta + \cos\theta$ gives $\sin\frac{\pi}{4} + \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2}$. Step 7: From the calculations, it's clear that the condition $\log 2 - 2\log\sec\theta = 2\log\cosec\theta - \log 2$ leads to $\sin 2\theta = 1$, and thus $\theta = 2n\pi + \frac{\pi}{4}$, with $\sin\theta + \cos\theta = \sqrt{2}$. This matches option (III), indicating that the correct combination is the one that corresponds to these values. Step 8: Therefore, the correct answer is the option that matches the condition $\theta = 2n\pi + \frac{\pi}{4}$ and $\sin\theta + \cos\theta = \sqrt{2}$, which is option 3. The final answer is $\boxed{3}$.