If `sec^4theta+sec^2theta=10+tan^4theta+tan^2theta", then "sin^2theta=`

Options

A. `2/3`
B. `3/4`
C. `4/5`
D. `5/6`

Explanation

To solve the equation \( \sec^4 \theta + \sec^2 \theta = 10 + \tan^4 \theta + \tan^2 \theta \) and find the value of \( \sin^2 \theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Rearranging the Equation**:
\[ \sec^4 \theta + \sec^2 \theta - \tan^4 \theta - \tan^2 \theta = 10 \]

2. **Using the Identity**:
We know that \( \sec^2 \theta - \tan^2 \theta = 1 \). Therefore, we can express \( \sec^4 \theta - \tan^4 \theta \) as:
\[ \sec^4 \theta - \tan^4 \theta = (\sec^2 \theta - \tan^2 \theta)(\sec^2 \theta + \tan^2 \theta) = 1 \cdot (\sec^2 \theta + \tan^2 \theta) \]
So, we can rewrite the equation as:
\[ \sec^2 \theta + \tan^2 \theta + 1 = 10 \]

3. **Simplifying Further**:
This simplifies to:
\[ \sec^2 \theta + \tan^2 \theta = 9 \]

4. **Substituting for \( \tan^2 \theta \)**:
Recall that \( \tan^2 \theta = \sec^2 \theta - 1 \). Substituting this into the equation gives:
\[ \sec^2 \theta + (\sec^2 \theta - 1) = 9 \]
This simplifies to:
\[ 2\sec^2 \theta - 1 = 9 \]

5. **Solving for \( \sec^2 \theta \)**:
Adding 1 to both sides:
\[ 2\sec^2 \theta = 10 \]
Dividing by 2:
\[ \sec^2 \theta = 5 \]

6. **Finding \( \cos^2 \theta \)**:
Since \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \), we have:
\[ \frac{1}{\cos^2 \theta} = 5 \implies \cos^2 \theta = \frac{1}{5} \]

7. **Finding \( \sin^2 \theta \)**:
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{1}{5} = \frac{4}{5} \]

### Final Answer:
\[ \sin^2 \theta = \frac{4}{5} \]