Mathematics

The key is to recognize that this sum has a telescoping structure hidden in the tangent and cotangent identities. Notice the doubling pattern in the angles ($\alpha, 2\alpha, 4\alpha, 8\alpha$) and coefficients ($1, 2, 4, 8$)—this suggests using the identity $\cot \theta - \tan \theta = 2\cot 2\theta$ repeatedly to collapse the left side. Work backwards from $\cot \alpha$ by expressing it in terms of larger angles, and you'll see how each term cancels strategically.

The key is to recognize that this sum has a telescoping structure hidden in the tangent and cotangent identities. Notice the doubling pattern in the angles ($\alpha, 2\alpha, 4\alpha, 8\alpha$) and coefficients ($1, 2, 4, 8$)—this suggests using the identity $\cot \theta - \tan \theta = 2\cot 2\theta$ repeatedly to collapse the left side. Work backwards from $\cot \alpha$ by expressing it in terms of larger angles, and you'll see how each term cancels strategically.