Cosine of multiples
The general formula for calculating the cosine of multiples was given by the French mathematician Viète
The latter term is alternately 1, 0, −1, 0, 1 and so on, so that half of the factors in the sum disappear.
Example 0
We know that cos (0) = 1 and see if the formula gives this also. We take n = 0 and get
The only term is
This result arises here because we have taken sin00 = 1.
Example 1
Now we want to see what happens with n = 1 and find
The two consecutive terms are then
so that only remains cos x = cos x.
Example 2
For n = 2 the formula yields the sum
The three consecutive terms are then
so that only remains
This is what we expected, because the addition formula for the cosine is
and as a direct result