If \tan A+\cot A=2, where 0 \lt A \lt 90^{\circ}, then what is the value of \tan^{2}A+\tan^{3}A+\tan^{4}A+....+\tan^{n}A?

Explanation

From \tan A + \frac{1}{\tan A} = 2, we get (\tan A - 1)^2 = 0, so \tan A = 1. The given expression is the sum of powers of 1 from 2 to n. The number of terms is (n - 2) + 1 = n - 1. Thus, the sum is n-1.

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