Let $A_{n}=\int_{0}^{\frac{\pi}{4}}tan^{n}x~dx,$ then $A_{10}+A_{8}=$

Question

Accepted Answer

To solve the problem, we need to evaluate the integrals $ A_{10} = \int_{0}^{\frac{\pi}{4}} \tan^{10} x \, dx $ and $ A_{8} = \int_{0}^{\frac{\pi}{4}} \tan^{8} x \, dx $, and then find their sum $ A_{10} + A_{8} $.

### Explanation:

The integral $ A_n = \int_{0}^{\frac{\pi}{4}} \tan^n x \, dx $ can be evaluated using a reduction formula. The reduction formula for the integral of powers of tangent is given by:

$$
A_n = \int_{0}^{\frac{\pi}{4}} \tan^n x \, dx = \frac{1}{n-1} - A_{n-2}
$$

This formula is valid for $ n \geq 2 $. We can use this formula to find $ A_{10} $ and $ A_{8} $.

#### Step-by-step Calculation:

1. **Calculate $ A_{10} $:**

Using the reduction formula:
   $$
   A_{10} = \frac{1}{9} - A_8
   $$

2. **Calculate $ A_{8} $:**

Similarly, using the reduction formula:
   $$
   A_8 = \frac{1}{7} - A_6
   $$

Continuing this process, we find:
   $$
   A_6 = \frac{1}{5} - A_4
   $$
   $$
   A_4 = \frac{1}{3} - A_2
   $$
   $$
   A_2 = \frac{1}{1} - A_0
   $$

3. **Calculate $ A_0 $:**

Since $ A_0 = \int_{0}^{\frac{\pi}{4}} \tan^0 x \, dx = \int_{0}^{\frac{\pi}{4}} 1 \, dx = \frac{\pi}{4} $.

4. **Substitute back to find $ A_2 $:**

$$
   A_2 = 1 - \frac{\pi}{4}
   $$

5. **Continue substituting back to find $ A_4, A_6, A_8 $:**

Substitute $ A_2 $ into the equation for $ A_4 $:
   $$
   A_4 = \frac{1}{3} - (1 - \frac{\pi}{4})
   $$

Continue this process to find $ A_6 $ and $ A_8 $.

6. **Calculate $ A_{10} + A_{8} $:**

Using the values obtained from the reduction formula, substitute back to find $ A_{10} $ and add it to $ A_8 $.

$$
   A_{10} + A_8 = \left(\frac{1}{9} - A_8\right) + A_8 = \frac{1}{9}
   $$

### Conclusion:

The sum $ A_{10} + A_{8} = \frac{1}{9} $, which corresponds to Option 2. This confirms that Option 2 is indeed the correct answer.

### Why Other Options are Incorrect:

- **Option 1 ($\frac{1}{8}$)**: This value does not match the calculated sum of $ A_{10} + A_{8} $ using the reduction formula.
- **Option 3 ($\frac{1}{11}$)**: This value is not consistent with the result obtained from the reduction formula.
- **Option 4 ($\frac{1}{13}$)**: This value is also not consistent with the calculated sum.

Thus, the only correct and consistent option is Option 2, $\frac{1}{9}$.