MHB How Do You Calculate the Sum of Squared Sines in Sequence?

[anemone]

Here is this week's POTW:

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Evaluate $\sin^2 x^\circ+\sin^2 (x+1)^\circ+\sin^2 (x+2)^\circ+\cdots+\sin^2 (x+179)^\circ$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solution::)

1. kaliprasad
2. greg1313
3. lfdahl

Solution of lfdahl:

Spoiler

\[\sum_{d=0}^{179}sin^2(x+d)^{\circ} \\\\=\sum_{d=0}^{89}(sin^2(x+d)^{\circ}+sin^2(x+d+90)^{\circ}) \\\\=\sum_{d=0}^{89}(sin^2(x+d)^{\circ}+cos^2(x+d)^{\circ}) \\\\ = \sum_{d=0}^{89}1 \\\\ = 90\]