`sin(90^circ+2theta)` is

Options

A. `sin 2theta`
B. `cos 2theta`
C. `- cos 2theta`
D. `- sin 2theta`

Explanation

To solve the problem \( \sin(90^\circ + 2\theta) \), we can use the sine addition formula. Here are the steps to find the solution:

### Step 1: Use the Sine Addition Formula
The sine addition formula states that:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
In our case, let \( A = 90^\circ \) and \( B = 2\theta \).

### Step 2: Substitute the Values
Substituting \( A \) and \( B \) into the formula, we get:
\[ \sin(90^\circ + 2\theta) = \sin(90^\circ) \cos(2\theta) + \cos(90^\circ) \sin(2\theta) \]

### Step 3: Evaluate the Sine and Cosine of 90 Degrees
We know that:
\[ \sin(90^\circ) = 1 \quad \text{and} \quad \cos(90^\circ) = 0 \]
Substituting these values into the equation gives:
\[ \sin(90^\circ + 2\theta) = 1 \cdot \cos(2\theta) + 0 \cdot \sin(2\theta) \]

### Step 4: Simplify the Expression
This simplifies to:
\[ \sin(90^\circ + 2\theta) = \cos(2\theta) \]

### Final Answer
Thus, the final result is:
\[ \sin(90^\circ + 2\theta) = \cos(2\theta) \]