Error 404 Followers: Sine Squared with a Tinge of Progressions, anyone?

The sum of 1-90 degrees of $\sin^2$ is,

$\sin^2{1^{\circ}}+\sin^2{2^{\circ}}+\ldots+\sin^2{45^{\circ}}+\ldots+\sin^2{88^{\circ}}+\sin^2{89^{\circ}}+\sin^2{90^{\circ}}$

but $\sin^2{90^{\circ}}=1$ and $\sin^2{45^{\circ}}=0.5$ that is,

$\sin^2{1^{\circ}}+\sin^2{2^{\circ}}+\ldots+0.5+\ldots+\sin^2{88^{\circ}}+\sin^2{89^{\circ}}+1$

$=1+0.5+\underbrace{[(\sin^2{1^{\circ}}+\sin^2{89^{\circ}})+(\sin^2{2^{\circ}}+\sin^2{88^{\circ}})+\ldots+(\sin^2{44^{\circ}}+\sin^2{46^{\circ}})]}_\text{44 groups}$

$=1.5+\underbrace{(1+1+1+\ldots)}_\text{44 numbers}=1.5+(44\times 1)=45.5$

The sum of 91-180 degrees of $\sin^2$ is,

$\sin^2{91^{\circ}}+\sin^2{92^{\circ}}+\ldots+\sin^2{135^{\circ}}+\ldots+\sin^2{178^{\circ}}+\sin^2{179^{\circ}}+\sin^2{180^{\circ}}$

but $\sin^2{180^{\circ}}=0$ and $\sin^2{135^{\circ}}=\sin^2{45^{\circ}}=0.5$ that is,

$\sin^2{91^{\circ}}+\sin^2{92^{\circ}}+\ldots+0.5+\ldots+\sin^2{178^{\circ}}+\sin^2{179^{\circ}}$

$=\sin^2{89^{\circ}}+\sin^2{88^{\circ}}+\ldots+0.5+\ldots+\sin^2{2^{\circ}}+\sin^2{1^{\circ}}$

$=0.5+\underbrace{[(\sin^2{89^{\circ}}+\sin^2{1^{\circ}})+(\sin^2{88^{\circ}}+\sin^2{2^{\circ}})+\ldots+(\sin^2{46^{\circ}}+\sin^2{44^{\circ}})]}_\text{44 groups}$

$=0.5+\underbrace{(1+1+1+\ldots)}_\text{44 numbers}=0.5+(44\times 1)=44.5$

The sum of 181-270 degrees of $\sin^2$ is,

$\sin^2{181^{\circ}}+\sin^2{182^{\circ}}+\ldots+\sin^2{225^{\circ}}+\ldots+\sin^2{268^{\circ}}+\sin^2{269^{\circ}}+\sin^2{270^{\circ}}$

but $\sin^2{270^{\circ}}=\sin^2{90^{\circ}}=1$ and $\sin^2{225^{\circ}}=\sin^2{45^{\circ}}=0.5$ that is,

$\sin^2{181^{\circ}}+\sin^2{182^{\circ}}+\ldots+0.5+\ldots+\sin^2{268^{\circ}}+\sin^2{269^{\circ}}+1$

$=\sin^2{1^{\circ}}+\sin^2{2^{\circ}}+\ldots+0.5+\ldots+\sin^2{88^{\circ}}+\sin^2{89^{\circ}}+1=45.5$

The sum of 271-360 degrees of $\sin^2$ is,

$\sin^2{271^{\circ}}+\sin^2{272^{\circ}}+\ldots+\sin^2{315^{\circ}}+\ldots+\sin^2{358^{\circ}}+\sin^2{359^{\circ}}+\sin^2{360^{\circ}}$

but $\sin^2{360^{\circ}}=\sin^2{180^{\circ}}=0$ and $\sin^2{315^{\circ}}=\sin^2{45^{\circ}}=0.5$ that is,

$\sin^2{271^{\circ}}+\sin^2{272^{\circ}}+\ldots+0.5+\ldots+\sin^2{358^{\circ}}+\sin^2{359^{\circ}}$

$=\sin^2{89^{\circ}}+\sin^2{88^{\circ}}+\ldots+0.5+\ldots+\sin^2{2^{\circ}}+\sin^2{1^{\circ}}=44.5$

Hence, the sum of 1-360 degrees of $\sin^2$ is, $45.5+44.5+45.5+44.5=90+90=\boxed{180}$

There's a better solution;

Convert in the form of cosines using half angle; using sin^2(x) = (1-cos(2x))/2

( 1-cos(2) ) /2+ ( 1-cos(4) ) /2+ ( 1-cos(6) ) /2+ ..........+( 1-cos(720) ) /2

bow use the formulae ; cos( a) + cos(a+b) + cos(a +2b) +.....+ cos(a +(n-1)b ) = (sin(nb/2) cos(a +((n-1)*b)/2))) /sin (b/2)

it will be equal to zero and the remaining (1/2) (s) will add upto 180