Prove that `2 sin 2^(@)+4sin 4^(@)+6sin 6^(@)+........+180sin 180^(@)=90 cot 1^(@)`.

Explanation

To prove that \( 2 \sin 2^\circ + 4 \sin 4^\circ + 6 \sin 6^\circ + \ldots + 180 \sin 180^\circ = 90 \cot 1^\circ \), we will follow a systematic approach.

### Step 1: Rewrite the Left-Hand Side
We start with the left-hand side:
\[ S = 2 \sin 2^\circ + 4 \sin 4^\circ + 6 \sin 6^\circ + \ldots + 180 \sin 180^\circ \]
Notice that the last term \( \sin 180^\circ = 0 \), so we can ignore it. The last non-zero term is \( 178 \sin 178^\circ \).
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