The number of solutions to $\sin \left(\pi \sin ^2 \theta\right)+\sin \left(\pi \cos ^2 \theta\right)=2 \cos \left(\frac{\pi}{2} \cos \theta\right)$ satisfying $0 \leq \theta \leq 2 \pi$ is
$1$
$2$
$4$
$7$
For each positive real number $\lambda$. Let $A_\lambda$ be the set of all natural numbers $n$ such that $|\sin (\sqrt{n+1})-\sin (\sqrt{n})|<\lambda$. Let $A_\lambda^c$ be the complement of $A_\lambda$ in the set of all natural numbers. Then,
If $(2\cos x - 1)(3 + 2\cos x) = 0,\,0 \le x \le 2\pi $, then $x = $
The number of solutions $x$ of the equation $\sin \left(x+x^2\right)-\sin \left(x^2\right)=\sin x$ in the interval $[2,3]$ is
If $\cos \theta + \sec \theta = \frac{5}{2}$, then the general value of $\theta $ is
If $3({\sec ^2}\theta + {\tan ^2}\theta ) = 5$, then the general value of $\theta $ is