If $$4 - 2 \sin ^2 \theta - 5 \cos \theta = 0, 0^\circ < \theta < 90^\circ$$, then the value of $$ \sin \theta + \tan \theta$$ is:

$$4 - 2 \sin ^2 \theta - 5 \cos \theta = 0$$

Using the equation ,

$$ \sin ^2 \theta + \cos ^2 \theta = 1$$ , in above equation ,

$$4 - 2 (1 - \cos ^2 \theta) - 5 \cos \theta = 0$$

$$4 - 2 +  2\cos ^2 \theta - 5 \cos \theta = 0$$

$$2\cos ^2 \theta - 5 \cos \theta +2 = 0$$

$$\cos\theta = \frac{5 \pm \sqrt{25 -16}}{4}$$

$$\cos\theta = \frac{5 \pm 3}{4}$$

$$\cos\theta = 2 or \frac{1}{2}$$

$$\cos\theta  can't  be  more  than 1, so \cos\theta=\frac{1}{2}$$

$$\sin\theta =  \frac{\sqrt{3}}{2}$$

$$\tan\theta = \sqrt{3}$$

$$ \sin \theta + \tan \theta = \frac{\sqrt{3}}{2} + \sqrt{3} = \frac{3\sqrt{3}}{2}$$

So , the answer would be Option b)$$\frac{3\sqrt{3}}{2}$$.