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compute cos^2 1 +cos^2 2 +cos^2 3 ... +cos^2 89 (in degrees)

 Dec 13, 2023
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Use the double-angle formula for cosine: cos 2α = 2cos^2 α - 1.

 

Rewrite each cos^2 term using the double-angle formula: cos^2 1 = (cos 2 * 1 + 1) / 2, cos^2 2 = (cos 2 * 2 + 1) / 2, ..., cos^2 89 = (cos 2 * 89 + 1) / 2.

 

Substitute the new expressions into the original sum: (cos 2 * 1 + 1) / 2 + (cos 2 * 2 + 1) / 2 + ... + (cos 2 * 89 + 1) / 2.

 

Notice that the terms with "cos 2 * α" cancel out in pairs due to the periodic nature of the cosine function.

 

The sum simplifies to 44 / 2 = 22.

 Dec 13, 2023

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