`2 cos (theta - pi //2) + 3 sin (theta + pi//2) - (3sin theta + 2 cos theta ) = `?
A. `cos theta - sin theta `
B. `sin theta - cos theta`
C. `sin theta + cos theta`
D. `cot theta - tan theta` .

Options

A. B
B. D
C. C
D. A

Answer

To solve the equation \( 2 \cos(\theta - \frac{\pi}{2}) + 3 \sin(\theta + \frac{\pi}{2}) - (3 \sin \theta + 2 \cos \theta) = ? \), we can simplify the left-hand side step by step.

### Step 1: Simplify \( \cos(\theta - \frac{\pi}{2}) \) and \( \sin(\theta + \frac{\pi}{2}) \)

Using the trigonometric identities:
- \( \cos(\theta - \frac{\pi}{2}) = \sin \theta \)
- \( \sin(\theta + \frac{\pi}{2}) = \cos \theta \)

We rewrite the equation:

\[ 2 \cos(\theta - \frac{\pi}{2}) = 2 \sin \theta \]
\[ 3 \sin(\theta + \frac{\pi}{2}) = 3 \cos \theta \]

So the equation becomes:

\[ 2 \sin \theta + 3 \cos \theta - (3 \sin \theta + 2 \cos \theta) \]

### Step 2: Distribute the negative sign

Now, distribute the negative sign across the terms in the parentheses:

\[ 2 \sin \theta + 3 \cos \theta - 3 \sin \theta - 2 \cos \theta \]

### Step 3: Combine like terms

Now combine the like terms:

- For \( \sin \theta \):
\[ 2 \sin \theta - 3 \sin \theta = -1 \sin \theta = -\sin \theta \]

- For \( \cos \theta \):
\[ 3 \cos \theta - 2 \cos \theta = 1 \cos \theta = \cos \theta \]

Putting it all together, we have:

\[ -\sin \theta + \cos \theta = \cos \theta - \sin \theta \]

### Final Result

Thus, the final simplified expression is:

\[ \cos \theta - \sin \theta \]

### Conclusion

The correct answer is option **A: \( \cos \theta - \sin \theta \)**.

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