The value of $$\dfrac{(\cos 9^\circ + \sin 81^\circ)(\sec 9^\circ + \cosec 81^\circ)}{\sin 56^\circ \sec 34^\circ + \cos 25^\circ \cosec 65^\circ}$$ is:

Given that,

$$\dfrac{(\cos 9^\circ + \sin 81^\circ)(\sec 9^\circ + \cosec 81^\circ)}{\sin 56^\circ \sec 34^\circ + \cos 25^\circ \cosec 65^\circ}$$
$$=\dfrac{(\cos 9^\circ + \cos (90^\circ-81^\circ))(\sec 9^\circ + \sec(90^\circ-81^\circ))}{\sin 56^\circ \cosec(90^\circ-34^\circ)  + \cos 25^\circ \sec(90^\circ-65^\circ)}$$

$$=\dfrac{(\cos 9^\circ + \cos (9^\circ))(\sec 9^\circ + \sec(9^\circ))}{\sin 56^\circ \cosec(56^\circ)  + \cos 25^\circ \sec(25^\circ)}$$

$$=\dfrac{(2\cos 9^\circ))(2\sec 9^\circ ))}{\dfrac{\sin 56^\circ}{\sin(56^\circ)} + \dfrac{\cos 25^\circ}{\sec(25^\circ})}$$

$$=\dfrac{4}{2}$$
$$=2$$