Different methods, different answers.

The left and right side aren't the same function, they just happen to intersect at a specific angle $\theta$. Because of that, you cannot expect both to have the same derivative, so Method 1 isn't correct.

Method $(2)$ is correct, and as you found, the correct choice is $(\text{d})$.

Method $(1)$ is seriously flawed.

The given equation does not assert that two functions of $\theta$ are identically equal. Rather it asks for a value of $\theta$ for which the two functions of $\theta$ are equal.

To dramatize the error, suppose we want a value of $\theta$ with $0 < \theta < \pi$ such that $$\sin(\theta) = \frac{\theta}{2}$$ It makes no sense to differentiate both sides. If you differentiated anyway, ignoring the fact that equation does not assert that the LHS and RHS are the same function, you would get $$\cos(\theta) = \frac{1}{2}$$ yielding $\theta = \pi/3$, which doesn't satisfy the original equation.