If `l_(n)=int_(0)^((pi)/(4)) tan^(n) xdx` show that `(1)/(l_(2)+l_(4)),(1)/(l_(3)+l_(5)),(1)/(l_(4)+l_(6)),(1)/(l_(5)+l_(7)),"...."` from an AP. Find its common difference.

Explanation

To solve the problem, we need to show that the sequence \( \frac{1}{l_2 + l_4}, \frac{1}{l_3 + l_5}, \frac{1}{l_4 + l_6}, \frac{1}{l_5 + l_7}, \ldots \) forms an arithmetic progression (AP) and find its common difference.

### Step 1: Define the integral \( l_n \)
We start with the definition of \( l_n \):
\[ l_n = \int_0^{\frac{\pi}{4}} \tan^n x \, dx \]

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