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 \( f(\sin^2 \theta) + f(\cos^2 \theta) = \tan^2 \theta \) where \( f(x) = \frac{25^x}{25^x + 5} \) and \( \theta \) is in the interval \([0, 10\pi]\), we will follow these steps:

### Step 1: Evaluate \( f(\sin^2 \theta) \) and \( f(\cos^2 \theta) \)

Given the function:
\[ f(x) = \frac{25^x}{25^x + 5} \]
We can express \( f(\sin^2 \theta) \) and \( f(\cos^2 \theta) \):

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

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

### Step 2: Combine \( f(\sin^2 \theta) \) and \( f(\cos^2 \theta) \)

Now, we need to find:
\[ f(\sin^2 \theta) + f(\cos^2 \theta) \]
This can be computed as:
\[ f(\sin^2 \theta) + f(\cos^2 \theta) = \frac{25^{\sin^2 \theta}}{25^{\sin^2 \theta} + 5} + \frac{25^{\cos^2 \theta}}{25^{\cos^2 \theta} + 5} \]

### Step 3: Simplify the expression

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express:
\[ f(\sin^2 \theta) + f(\cos^2 \theta) = \frac{25^{\sin^2 \theta}(25^{\cos^2 \theta} + 5) + 25^{\cos^2 \theta}(25^{\sin^2 \theta} + 5)}{(25^{\sin^2 \theta} + 5)(25^{\cos^2 \theta} + 5)} \]

### Step 4: Set the equation equal to \( \tan^2 \theta \)

We need to solve:
\[ f(\sin^2 \theta) + f(\cos^2 \theta) = \tan^2 \theta \]

### Step 5: Analyze the behavior of \( \tan^2 \theta \)

The function \( \tan^2 \theta \) has periodic solutions. Specifically, \( \tan^2 \theta = 1 \) at:
\[ \theta = \frac{\pi}{4} + n\pi \quad \text{for } n \in \mathbb{Z} \]
Within the interval \([0, 10\pi]\), we can find the number of solutions.

### Step 6: Count the solutions

The solutions for \( \tan^2 \theta = 1 \) in the interval \([0, 10\pi]\) can be determined by counting the number of times \( \theta \) hits \( \frac{\pi}{4} + n\pi \):

1. The first solution is at \( \frac{\pi}{4} \).
2. The subsequent solutions are spaced by \( \pi \).

Thus, the solutions are:
\[ \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}, \ldots, \frac{39\pi}{4} \]
To find the total number of solutions, we can calculate how many \( n \) values fit in the range:
\[ 0 \leq \frac{\pi}{4} + n\pi \leq 10\pi \]
This gives:
\[ 0 \leq n \leq 39 \]
Thus, \( n \) can take values from \( 0 \) to \( 39 \), giving us \( 40 \) solutions.

### Conclusion

The total number of solutions of the equation \( f(\sin^2 \theta) + f(\cos^2 \theta) = \tan^2 \theta \) in the interval \([0, 10\pi]\) is \( 40 \).