1 + cos(10y)=2cos^2(5y)

1 + cos(10y)=2cos^2(5y)

Remember the identy for cos(2a):

cos(2x) = cos^2(a) - sin^2(a).

And if you apply sin^2(a) + cos^2(a) = 1, you can turn this either:

cos(2a) = 1-2sin^2(a), or
cos(2a) = 2 cos^2(a) -1

So you actually have three forms of identity for cos(2a). If you consider that your problem is asking about cos(10y), and that 10y = 2 x 5y, then you have

1 + cos(2 x 5y).

Try using the various forms of identities for cos(2a) with the above, and see if you can work it out.