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

$$3 \sin \theta = 2 \cos^2 \theta$$

$$\frac{\sin\theta}{\cos^2\theta}=\frac{2}{3}$$

Where $$0^\circ < \theta < 90^\circ$$

So $$\theta = 30^\circ, 45^\circ and 60^\circ$$

When $$\theta = 30^\circ$$

$$\frac{\sin 30^\circ}{\cos^2\ 30^\circ}=\frac{2}{3}$$

$$\frac{\frac{1}{2}}{\left(\frac{\sqrt{\ 3}}{2}\right)^2}=\frac{2}{3}$$

$$\frac{\frac{1}{2}}{\frac{3}{4}}=\frac{2}{3}$$

$$\frac{2}{3}=\frac{2}{3}$$

Here $$\theta = 30^\circ$$ is satisfying the given equation. So we need not to check other.

Value of $$(\tan \theta + \cos \theta + \sin \theta)$$ = $$(\tan 30^\circ + \cos 30^\circ + \sin 30^\circ)$$

= $$\left(\frac{1}{\sqrt{\ 3}}+\frac{\sqrt{\ 3\ }}{2}+\frac{1}{2})\right)$$

= $$\left(\frac{1}{\sqrt{\ 3}}+\frac{\sqrt{\ 3\ }\ +1}{2}\right)$$

= $$\left(\frac{2+\sqrt{\ 3\ }\left(\sqrt{\ 3\ }\ +1\right)}{2\sqrt{\ 3\ }}\right)$$

= $$\left(\frac{2+3+\sqrt{\ 3\ }}{2\sqrt{\ 3\ }}\right)$$

= $$\left(\frac{5+\sqrt{\ 3\ }}{2\sqrt{\ 3\ }}\right)$$

Multiply $$\sqrt{\ 3\ }$$ in numerator and denominator.

= $$\left(\frac{5+\sqrt{\ 3\ }}{2\sqrt{\ 3\ }}\right)\times\ \frac{\sqrt{\ 3\ }}{\sqrt{\ 3\ }}$$

= $$\left(\frac{5\sqrt{\ 3\ }+3}{2\times\ 3}\right)$$

= $$\left(\frac{5\sqrt{\ 3\ }+3}{6}\right)$$