The value of $$\frac{(\cos 9^\circ + \sin 81^\circ)(\sec 9^\circ + \cosec 81^\circ)}{2 \sin^2 63^\circ + 1 + 2 \sin^2 27^\circ}$$ is:

$$\frac{(\cos 9^\circ + \sin 81^\circ)(\sec 9^\circ + \cosec 81^\circ)}{2 \sin^2 63^\circ + 1 + 2 \sin^2 27^\circ}$$

= $$\frac{(\cos(90 - 81) + \sin 81^\circ)(\sec (90 - 81) + \cosec 81^\circ)}{2 \sin^2 63^\circ + 1 + 2 \sin^2(90 - 63)}$$

= $$\frac{(\sin 81^\circ + \sin 81^\circ)(\cosec 81^\circ +\cosec 81^\circ)}{2 \sin^2 63^\circ + 1 + 2 \cos^2 63^\circ}$$

= $$\frac{(2\sin 81^\circ)(2\cosec 81^\circ}{2 + 1}$$ = $$\frac{4}{3}$$