`tanthetasin((pi)/(2)+theta)cos((pi)/(2)-theta)=?`

Options

A. 1
B. 0
C. `(1)/(sqrt(2))`
D. `sin^(2)theta`

Answer

To solve the expression \( \tan \theta \sin\left(\frac{\pi}{2} + \theta\right) \cos\left(\frac{\pi}{2} - \theta\right) \), we can follow these steps:

### Step 1: Rewrite the trigonometric functions
We know the following trigonometric identities:
- \( \sin\left(\frac{\pi}{2} + \theta\right) = \cos \theta \)
- \( \cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta \)

Using these identities, we can rewrite the expression:
\[ \tan \theta \sin\left(\frac{\pi}{2} + \theta\right) \cos\left(\frac{\pi}{2} - \theta\right) = \tan \theta \cdot \cos \theta \cdot \sin \theta \]

### Step 2: Substitute \( \tan \theta \)
Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). We substitute this into the expression:
\[ \tan \theta \cdot \cos \theta \cdot \sin \theta = \frac{\sin \theta}{\cos \theta} \cdot \cos \theta \cdot \sin \theta \]

### Step 3: Simplify the expression
The \( \cos \theta \) in the numerator and denominator cancels out:
\[ \frac{\sin \theta \cdot \cos \theta \cdot \sin \theta}{\cos \theta} = \sin^2 \theta \]

### Conclusion
Thus, the final simplified expression is:
\[ \tan \theta \sin\left(\frac{\pi}{2} + \theta\right) \cos\left(\frac{\pi}{2} - \theta\right) = \sin^2 \theta \]

### Final Answer
\[ \sin^2 \theta \]

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