If `cot theta =7/8`
Find the value of `sin^(2) theta +cos^(2) theta ` .

Options

A. 1
B. 0
C. `1/2`
D. `1/3`

Answer

To solve the problem where we need to find the value of \( \sin^2 \theta + \cos^2 \theta \) given that \( \cot \theta = \frac{7}{8} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Cotangent**:
\[ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{7}{8} \]
Here, we can assign the lengths of the sides of the triangle:
- Adjacent side = 7
- Opposite side = 8

2. **Finding the Hypotenuse**:
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2 \]
\[ \text{Hypotenuse}^2 = 7^2 + 8^2 = 49 + 64 = 113 \]
Therefore, the hypotenuse is:
\[ \text{Hypotenuse} = \sqrt{113} \]

3. **Calculating Sine and Cosine**:
Now we can find \( \sin \theta \) and \( \cos \theta \):
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{8}{\sqrt{113}} \]
\[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{7}{\sqrt{113}} \]

4. **Finding \( \sin^2 \theta + \cos^2 \theta \)**:
Now we calculate \( \sin^2 \theta + \cos^2 \theta \):
\[ \sin^2 \theta = \left(\frac{8}{\sqrt{113}}\right)^2 = \frac{64}{113} \]
\[ \cos^2 \theta = \left(\frac{7}{\sqrt{113}}\right)^2 = \frac{49}{113} \]
Adding these two:
\[ \sin^2 \theta + \cos^2 \theta = \frac{64}{113} + \frac{49}{113} = \frac{64 + 49}{113} = \frac{113}{113} = 1 \]

### Final Answer:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]