If $$3(\cot^2 \phi - \cos \phi) = \cos^2 \phi, 0^\circ < \phi < 90^\circ$$, then the value of $$(\tan^2 \phi + \cosec^2 \phi + \sin^2 \phi)$$ is:

$$3(\cot^2 \phi - \cos \phi) = \cos^2 \phi$$
Put the value of $$\phi = 60\degree$$,
$$3(\cot^2 60\degree - \cos 60\degree) = \cos^2 60\degree$$
$$3(\frac{1}{3} - \frac{1}{4}) = \frac{1}{4}$$
$$3(\frac{1}{12}) = \frac{1}{4}$$
$$\frac{1}{4} = \frac{1}{4}$$
Now,
$$(\tan^2 \phi + \cosec^2 \phi + \sin^2 \phi)$$
Put the value of $$\phi = 60\degree$$,
= $$(\tan^260\degree + \cosec^260\degree + \sin^2 60\degree)$$
= $$3 + \frac{4}{3} + \frac{3}{4}$$
= $$\frac{36 + 16 + 9}{12} = \frac{61}{12}$$