1)
$\sin^4x+\cos^4x$
$=\left(\sin^2x+\cos^2x\right)^2-2\sin^2x\cos^2x$
$=1^2-\dfrac{1}{2}.\left(2\sin x\cos x\right)^2$
$=1-\dfrac{1}{2}.\sin^22x$
$=1-\dfrac{1}{2}.\dfrac{1-\cos4x}{2}$
$=\dfrac{3}{4}+\dfrac{1}{4}\cos4x$
2)
$\dfrac{1}{\sin2x}+\cot2x$
$=\dfrac{1}{\sin2x}+\dfrac{\cos2x}{\sin2x}$
$=\dfrac{\cos2x+1}{\sin2x}$
$=\dfrac{2\cos^2x}{2\sin x\cos x}$
$=\dfrac{\cos x}{\sin x}$
$=\cot x$
3)
$\cos3x.\sin^3x+cos^3x.\sin3x$
$=\sin^3x(4\cos^3x-3\cos x)+\cos^3x(3\sin x-4\sin^3x)$
$=4\sin^3x\cos^3x-3\sin^3x\cos x-4\sin^3x\cos^3x+3\sin x\cos^3x$
$=3\sin x\cos x(\cos^2x-\sin^2x)$
$=3\sin x\cos x\cos2x$
$=\dfrac{3}{2}\sin2x\cos2x$
$=\dfrac{3}{4}\sin4x$