The number of real solutions $x$ of the equation $\cos ^2(x \sin (2 x))+\frac{1}{1+x^2}=\cos ^2 x+\sec ^2 x$ is

If $\cos \theta = - \frac{1}{{\sqrt 2 }}$ and $\tan \theta = 1$, then the general value of $\theta $ is

Let $f(x) = \cos \sqrt {x,} $ then which of the following is true

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 $\cos \theta + \sec \theta = \frac{5}{2}$, then the general value of $\theta $ is

If $1 + \cot \theta = {\rm{cosec}}\theta $, then the general value of $\theta $ is