For any real $\theta$, the maximum value of ${\cos }^2\left(\cos \theta \right)+{\sin }^2\left(\sin \theta \right)$ is

If $\alpha ,\beta (\alpha <\beta )$ be the points where the function

$f\left(x\right)=\underset{-1}{\overset{x}{\int }}t(e^t-1)(3t-1){(t-2)}^{3}dt$ has local minimum and $y=\frac{{\sec }^2\theta -\tan \theta }{{\sec }^2\theta +\tan \theta }$,then

If $k$ be the value of $x$ at which the function $f\left(x\right)=\underset{-1}{\overset{x}{\int }}t(e^t-1)(t-1){(t-2)}^3{(t-3)}^{5}dt$ has maximum value and $\sin x+\cos ecx=k,$ then for $n\in N,{\sin }^{n}x+\cos e{c}^{n}x=$

Let $A=\sin x+\tan x$ and $B=2x$ in the interval $0<x<\frac{\pi }{2}$ then