The value of `(sintheta+costheta)^(2)`=
A. `1+sin^(2)theta`
B. `sin^(2)theta+cos^(2)theta`
C. `1+2costhetasintheta`
D. `cos^(2)theta+1`

Options

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

Answer

To solve the expression \((\sin \theta + \cos \theta)^2\), we can use the algebraic identity for the square of a binomial:

\[ (a + b)^2 = a^2 + b^2 + 2ab \]

### Step 1: Identify \(a\) and \(b\)
Here, let \(a = \sin \theta\) and \(b = \cos \theta\).

### Step 2: Apply the formula
Using the formula, we can expand \((\sin \theta + \cos \theta)^2\):

\[ (\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta \]

### Step 3: Simplify using Pythagorean identity
We know from the Pythagorean identity that:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

Substituting this into our expression gives:

\[ (\sin \theta + \cos \theta)^2 = 1 + 2 \sin \theta \cos \theta \]

### Step 4: Final expression
Thus, we can conclude that:

\[ (\sin \theta + \cos \theta)^2 = 1 + 2 \sin \theta \cos \theta \]

### Step 5: Identify the correct option
Looking at the options provided:
- A. \(1 + \sin^2 \theta\)
- B. \(\sin^2 \theta + \cos^2 \theta\)
- C. \(1 + 2 \cos \theta \sin \theta\)
- D. \(\cos^2 \theta + 1\)

The correct answer is **C. \(1 + 2 \sin \theta \cos \theta\)**.