If cospΘ + cosqΘ = o. prove that the different values of Θ form two arithmetical progressions in which the common differences are 2π/p+q and 2π/p-q respectively.

Answer 1

cos px = -cos qx

cos px = cos ( pi - qx )

px = n ( pi ) + / - qx

taking the first case

px = n ( pi ) + qx

x ( p - q ) = n ( pi )

x = n ( ? p i) / ( p - q )

taking the second case we get

x = n ( ? pi ) / ( p + q )

We find the common difference is what is asked.