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sum \sqrt{1+k^2} sin(a_k-a_{1000}) (VI Soros Olympiad 1990-00 R1 9.8)

Source:

May 27, 2024

algebratrigonometrySum

Problem Statement

Let

a_n

denote an angle from the interval for each

\left( 0, \frac{\pi}{2}\right)

, the tangent of which is equal to

n

. Prove that

\sqrt{1+1^2} \sin(a_1-a_{1000}) + \sqrt{1+2^2} \sin(a_2-a_{1000})+...+\sqrt{1+2000^2} \sin(a_{2000}-a_{1000}) = \sin a_{1000}