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Let Us Divide The Question Into Two Parts
First Let Us Solve For Cosine Series
cos1. cos2. cos3 ..... cos89
Let Us Club The First And Last Terms,then second and second last and so on
(cos1. cos89) (cos2. cos88) (cos3 .cos87) .... (cos44. cos46) cos45
cos45 will remain as no pairing will occur
Now
cos(90-x) = sinx
Applying The Above formula In the series
(cos1. sin1) (cos2. sin2) .... (cos44.sin44). cos45
Similarly Doing For Sine Series
(sin1. sin89) (sin2. sin88) (sin3. sin87) .... (sin44. sin46). sin45
Applying sin(90-x) = cosx
(sin1. cos1) (sin2. cos2) (sin3. cos3) .... (sin44. cos44). sin45
Now We Know That Sin45 = Cos45
So The Final Sine Series Becomes
(cos1.sin1) (cos2.sin2) (cos3.sin3) .... (cos44. sin44). cos45
Which is exactly the same as cosine series
They are exactly same series
so if we subtraction them
Answer= Zero
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