Question

`Sin^2(π/12)+sin^2(3π)/(12)+sin^2(5π)/(12)+sin^2(7π)/(12)+sin^2(9π)/(12)+sin^2(11π)/(12)=?`

Answer 1

`sin^2(pi/12)+sin^2((3pi)/12)+sin^2((5pi)/12)+sin^2((7pi)/12)+sin^2((9pi)/12)+sin^2((11π)/12)=3`

Explanation:

Observe that if `sin(pi-A)=sinA`, hence `sin^2((11pi)/12)=sin^2(pi-pi/12)=sin^2(pi/12)` and similarly `sin^2((9pi)/12)=sin^2((3pi)/12)` and `sin^2((7pi)/12)=sin^2((5pi)/12)` and

`sin^2(pi/12)+sin^2((3pi)/12)+sin^2((5pi)/12)+sin^2((7pi)/12)+sin^2((9pi)/12)+sin^2((11π)/12)`

= `2sin^2(pi/12)+2sin^2((3pi)/12)+2sin^2((5pi)/12)`

Now as `2sin^2B=1-cos2B`, the above is

`3-[cos(pi/6)+cos(pi/2)+cos((5pi)/6)]`

= `3-[cos(pi/6)+0+cos(pi-pi/6)]`

= `3-[cos(pi/6)-cos(pi/6)]`

= `3`