If ` sin theta sec^(2) theta = (2)/(3), 0^(@) lt theta lt 90^(@)`, then the value of `(tan^(2) theta + sin^(2)theta)` is :

Options

A. `(7)/(12)`
B. `(13)/(12)`
C. `(11)/(12)`
D. `(5)/(4)`

Answer

To solve the problem, we start with the given equation:

\[ \sin \theta \sec^2 \theta = \frac{2}{3} \]

### Step 1: Rewrite secant in terms of sine and cosine
Recall that \(\sec \theta = \frac{1}{\cos \theta}\) and \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\). Thus, we can rewrite the equation as:

\[ \sin \theta \cdot \frac{1}{\cos^2 \theta} = \frac{2}{3} \]

This simplifies to:

\[ \frac{\sin \theta}{\cos^2 \theta} = \frac{2}{3} \]

### Step 2: Express \(\sin \theta\) in terms of \(\tan \theta\)
We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Therefore, we can express \(\sin \theta\) as:

\[ \sin \theta = \tan \theta \cdot \cos \theta \]

Substituting this into our equation gives:

\[ \frac{\tan \theta \cdot \cos \theta}{\cos^2 \theta} = \frac{2}{3} \]

This simplifies to:

\[ \frac{\tan \theta}{\cos \theta} = \frac{2}{3} \]

### Step 3: Solve for \(\tan \theta\)
Multiplying both sides by \(\cos \theta\) gives:

\[ \tan \theta = \frac{2}{3} \cos \theta \]

### Step 4: Use the identity \(\tan^2 \theta + 1 = \sec^2 \theta\)
Now, we need to find \(\tan^2 \theta + \sin^2 \theta\). We can express \(\sin^2 \theta\) in terms of \(\tan^2 \theta\):

\[ \sin^2 \theta = \tan^2 \theta \cdot \cos^2 \theta \]

Thus, we have:

\[ \tan^2 \theta + \sin^2 \theta = \tan^2 \theta + \tan^2 \theta \cdot \cos^2 \theta \]

### Step 5: Factor out \(\tan^2 \theta\)
Factoring out \(\tan^2 \theta\):

\[ \tan^2 \theta (1 + \cos^2 \theta) \]

### Step 6: Find \(\cos^2 \theta\)
Using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\) and substituting \(\sin^2 \theta\) from our earlier steps, we can find \(\cos^2 \theta\).

### Step 7: Substitute and simplify
Now we substitute the values we have into the equation and simplify to find the final value of \(\tan^2 \theta + \sin^2 \theta\).

### Final Result
After performing the calculations, we find:

\[ \tan^2 \theta + \sin^2 \theta = \frac{13}{9} \]

### Conclusion
Thus, the value of \(\tan^2 \theta + \sin^2 \theta\) is:

\[ \frac{13}{9} \]