# Verify for any integer n: \cos\frac{(2n-1)\pi}{2}=0. Be sure to include a

Verify for any integer n: $$\displaystyle{\cos{{\frac{{{\left({2}{n}-{1}\right)}\pi}}{{{2}}}}}}={0}$$. Be sure to include an explanation of potential cases that may exist with the value of n.

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SchulzD
This is question related trigonometry basic problem
$$\displaystyle{\cos{{\frac{{{\left({2}{n}-{1}\right)}\pi}}{{{2}}}}}}$$
$$\displaystyle{\cos{{\left({\frac{{{2}{n}}}{{{2}}}}-{\frac{{{1}}}{{{2}}}}\right)}}}\pi$$
$$\displaystyle\Rightarrow{\cos{{\left({n}\pi-{\frac{{\pi}}{{{2}}}}\right)}}}$$
if n is even
then $$\displaystyle{\cos{{\left({n}\pi-{0}\right)}}}={\cos{\theta}}$$
then $$\displaystyle{\cos{{\left({n}\pi-{\frac{{\pi}}{{{2}}}}\right)}}}={\cos{{\left({\frac{{\pi}}{{{2}}}}\right)}}}={0}$$
if n is odd
then $$\displaystyle{\cos{{\left({n}\pi-\theta\right)}}}=-{\cos{\theta}}$$
hence $$\displaystyle{\cos{{\left({n}\pi-{\frac{{\pi}}{{{2}}}}\right)}}}=-{\cos{{\frac{{\pi}}{{{2}}}}}}={0}$$
hence we can say that
$$\displaystyle{\cos{{\left({\frac{{{2}{n}-{1}}}{{{2}}}}\right)}}}\pi={0}$$ for all integral values of 'n'" на "This is question related trigonometry basic problem
$$\displaystyle{\cos{{\frac{{{\left({2}{n}-{1}\right)}\pi}}{{{2}}}}}}$$
$$\displaystyle{\cos{{\left({\frac{{{2}{n}}}{{{2}}}}-{\frac{{{1}}}{{{2}}}}\right)}}}\pi$$
$$\displaystyle\Rightarrow{\cos{{\left({n}\pi-{\frac{{\pi}}{{{2}}}}\right)}}}$$
if n is even
then $$\displaystyle{\cos{{\left({n}\pi-{0}\right)}}}={\cos{\theta}}$$
then $$\displaystyle{\cos{{\left({n}\pi-{\frac{{\pi}}{{{2}}}}\right)}}}={\cos{{\left({\frac{{\pi}}{{{2}}}}\right)}}}={0}$$
if n is odd
then $$\displaystyle{\cos{{\left({n}\pi-\theta\right)}}}=-{\cos{\theta}}$$
hence $$\displaystyle{\cos{{\left({n}\pi-{\frac{{\pi}}{{{2}}}}\right)}}}=-{\cos{{\frac{{\pi}}{{{2}}}}}}={0}$$
hence we can say that
$$\displaystyle{\cos{{\left({\frac{{{2}{n}-{1}}}{{{2}}}}\right)}}}\pi={0}$$ for all integral values of 'n'
Verify for some integral value of "n"
for n=1 $$\displaystyle{\cos{{\left(\pi-{\frac{{\pi}}{{{2}}}}\right)}}}={\cos{{\left({\frac{{\pi}}{{{2}}}}\right)}}}={0}$$
for n=2 $$\displaystyle{\cos{{\left({2}\pi-{\frac{{\pi}}{{{2}}}}\right)}}}={\cos{{\left({\frac{{{3}\pi}}{{{2}}}}\right)}}}={0}$$
for n=3 $$\displaystyle{\cos{{\left({3}\pi-{\frac{{\pi}}{{{2}}}}\right)}}}=-{\cos{{\left({\frac{{\pi}}{{{2}}}}\right)}}}={0}$$
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