Let `f(x)=(25^(x))/(25^(x)+5)`, then the number of solution (s) of the equation `f(sin^(2)theta)+f(cos^(2)theta)=tan^(2)theta, theta` is/are `in[0, 10pi]`

Options

A. 10
B. 2
C. 40
D. 20

Answer

To solve the equation \( f(\sin^2 \theta) + f(\cos^2 \theta) = \tan^2 \theta \), we start by analyzing the function \( f(x) \):

1. **Define the function \( f(x) \)**:
\[ f(x) = \frac{25^x}{25^x + 5} \]

2. **Calculate \( f(\sin^2 \theta) \)**:
\[ f(\sin^2 \theta) = \frac{25^{\sin^2 \theta}}{25^{\sin^2 \theta} + 5} \]

3. **Calculate \( f(\cos^2 \theta) \)**:
\[ f(\cos^2 \theta) = \frac{25^{\cos^2 \theta}}{25^{\cos^2 \theta} + 5} \]

4. **Combine \( f(\sin^2 \theta) \) and \( f(\cos^2 \theta) \)**:
\[ f(\sin^2 \theta) + f(\cos^2 \theta) = \frac{25^{\sin^2 \theta}}{25^{\sin^2 \theta} + 5} + \frac{25^{\cos^2 \theta}}{25^{\cos^2 \theta} + 5} \]

5. **Simplify the expression**:
Let \( a = 25^{\sin^2 \theta} \) and \( b = 25^{\cos^2 \theta} \). Then:
\[ f(\sin^2 \theta) + f(\cos^2 \theta) = \frac{a}{a + 5} + \frac{b}{b + 5} \]

6. **Set the equation equal to \( \tan^2 \theta \)**:
\[ \frac{a}{a + 5} + \frac{b}{b + 5} = \tan^2 \theta \]

7. **Use the identity \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \)**:
\[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \]

8. **Find the values of \( \theta \)**:
We need to find the values of \( \theta \) in the interval \( [0, 10\pi] \) where the equation holds true. The solutions will occur at specific angles where \( \tan^2 \theta \) takes on certain values.

9. **Identify solutions**:
The solutions for \( \tan^2 \theta = 1 \) occur at:
\[ \theta = \frac{\pi}{4} + n\pi \quad \text{for } n \in \mathbb{Z} \]
In the interval \( [0, 10\pi] \), we find:
- \( \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}, \frac{13\pi}{4}, \frac{17\pi}{4}, \frac{21\pi}{4}, \frac{25\pi}{4}, \frac{29\pi}{4}, \frac{33\pi}{4}, \frac{37\pi}{4} \)

10. **Count the solutions**:
There are 10 solutions in total.

### Final Answer:
The number of solutions \( s \) of the equation \( f(\sin^2 \theta) + f(\cos^2 \theta) = \tan^2 \theta \) in the interval \( [0, 10\pi] \) is **10**.