If `xcostheta-sintheta=1`, then `x^(2)+(1+x^(2))sintheta` equals-

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the problem step by step, we start with the equation given:

**Step 1: Write down the given equation.**
Given:
\[ x \cos \theta - \sin \theta = 1 \]

**Step 2: Rearrange the equation to isolate \( x \cos \theta \).**
From the given equation, we can rearrange it as:
\[ x \cos \theta = 1 + \sin \theta \]

**Step 3: Choose a value for \( \theta \) to simplify calculations.**
To make calculations easier, we can choose \( \theta = 0 \).

**Step 4: Substitute \( \theta = 0 \) into the equation.**
When \( \theta = 0 \):
- \( \cos 0 = 1 \)
- \( \sin 0 = 0 \)

Substituting these values into the rearranged equation:
\[ x \cdot 1 = 1 + 0 \]
This simplifies to:
\[ x = 1 \]

**Step 5: Substitute \( x \) and \( \theta \) into the expression we need to evaluate.**
We need to evaluate:
\[ x^2 + (1 + x^2) \sin \theta \]
Substituting \( x = 1 \) and \( \theta = 0 \):
\[ 1^2 + (1 + 1^2) \sin 0 \]

**Step 6: Calculate the expression.**
Now, calculate each part:
- \( 1^2 = 1 \)
- \( 1 + 1^2 = 1 + 1 = 2 \)
- \( \sin 0 = 0 \)

Putting it all together:
\[ 1 + 2 \cdot 0 = 1 + 0 = 1 \]

Thus, the final answer is:
\[ \boxed{1} \]

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