What is the value of Cos0 x cos1 x cos2 x...........x cos89 = ?

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