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Question Number 144998 by imjagoll last updated on 01/Jul/21

$$\cos {}^{2}1°+\cos {}^{2}2°+\cos {}^{2}3°+...+\cos {}^{2}360°=?$$

Answered by Dwaipayan Shikari last updated on 01/Jul/21

$$\frac{360}{2}+\frac{1}{2}(cos2°+cos4°+cos6°+...+cos720°)$$$$=180+\frac{1}{2\sin \left(1°\right)}(sin3°-sin1°+sin5°-sin3°+...+sin\left(721°\right)-sin\left(719°\right))$$$$=180+\frac{1}{2\sin \left(1°\right)}(sin721°-sin1°)\sin \left(721°\right)=\sin \left(1°\right)$$$$=180+0=180\because {cos}^{2}n°=\frac{1+cos2n°}{2}$$

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