What is the value of $\frac{1}{2}$?

As $\theta$ increases from $0^{\circ}$ to $90^{\circ}$,the value of $\cos \theta$ :-

If $y = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!}$,then $\frac{dy}{dx} = $

The greatest value of the function $-5\, \sin\, \theta + 12\, \cos\, \theta$ is

Which of the following relation$(s)$ is/are correct?
$(A)\ \sin (90^{\circ}+\theta)=\cos (-\theta)$
$(B)\ \sin (180^{\circ}-\theta)=\cos (90^{\circ}-\theta)$
$(C)\ \sin (360^{\circ}-\theta)=\cos (360^{\circ}-\theta)$
$(D)\ \sin (180^{\circ}+\theta)=\cos (90^{\circ}-\theta)$

$\int\limits_0^{\frac{\pi }{4}} {\sin 2x} \,dx$ is equal to