The value of sum_{r=1}^{18} cos^2(5r)^circ,wh | vedclass

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(C) The given sum is $S = \sum_{r=1}^{18} \cos^2(5r)^\circ = \cos^2 5^\circ + \cos^2 10^\circ + \dots + \cos^2 85^\circ + \cos^2 90^\circ$. Since $\co

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$\frac{17}{2}$

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(C) The given sum is $S = \sum_{r=1}^{18} \cos^2(5r)^\circ = \cos^2 5^\circ + \cos^2 10^\circ + \dots + \cos^2 85^\circ + \cos^2 90^\circ$. Since $\cos 90^\circ = 0$,the last term is $0$. We can pair the terms using the identity $\cos^2 	heta + \cos^2(90^\circ - 	heta) = \cos^2 	heta + \sin^2 	heta = 1$. There are $17$ non-zero terms: $\cos^2 5^\circ, \cos^2 10^\circ, \dots, \cos^2 85^\circ$. Pairing them: $(\cos^2 5^\circ + \cos^2 85^\circ) + (\cos^2 10^\circ + \cos^2 80^\circ) + \dots + (\cos^2 40^\circ + \cos^2 50^\circ) + \cos^2 45^\circ$. There are $8$ such pairs,each summing to $1$,plus the middle term $\cos^2 45^\circ = (\frac{1}{\sqrt{2}})^2 = \frac{1}{2}$. Thus,$S = 8(1) + \frac{1}{2} = \frac{17}{2}$.

The value of $\sum_{r=1}^{18} \cos^2(5r)^\circ$,where $x^\circ$ denotes the $x$ degree,is equal to

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