`tan^(2)theta sin^(2) theta ` si equal to

Options

A. `tan^(2) theta - sin(2) theta`
B. `tan^(2) theta+ sin^(2) theta`
C. `(tan theta)/(sin^(2) theta)`
D. `sin^(2) theta cot^(2) theta`

Explanation

To solve the expression \( \tan^2 \theta \sin^2 \theta \), we can follow these steps:

### Step 1: Write the definition of \( \tan \theta \)
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Thus,
\[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \]

### Step 2: Substitute \( \tan^2 \theta \) into the expression
Now, we can substitute \( \tan^2 \theta \) into the original expression:
\[ \tan^2 \theta \sin^2 \theta = \left(\frac{\sin^2 \theta}{\cos^2 \theta}\right) \sin^2 \theta \]

### Step 3: Simplify the expression
This simplifies to:
\[ \frac{\sin^4 \theta}{\cos^2 \theta} \]

### Step 4: Rewrite \( \sin^4 \theta \)
We can express \( \sin^4 \theta \) in terms of \( \cos^2 \theta \):
\[ \sin^4 \theta = \sin^2 \theta \cdot \sin^2 \theta = \sin^2 \theta (1 - \cos^2 \theta) \]
Using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), we can rewrite:
\[ \sin^4 \theta = \sin^2 \theta (1 - \cos^2 \theta) \]

### Step 5: Substitute back into the expression
Now substitute this back into our expression:
\[ \tan^2 \theta \sin^2 \theta = \frac{\sin^2 \theta (1 - \cos^2 \theta)}{\cos^2 \theta} \]

### Step 6: Split the fraction
This can be split into two parts:
\[ = \frac{\sin^2 \theta}{\cos^2 \theta} - \frac{\sin^2 \theta \cos^2 \theta}{\cos^2 \theta} \]
\[ = \tan^2 \theta - \sin^2 \theta \]

### Final Result
Thus, we have:
\[ \tan^2 \theta \sin^2 \theta = \tan^2 \theta - \sin^2 \theta \]