M2 - Circular Motion - non-horizontal forces

I'm doing question 6a

Here are my workings. They're not right as the answer in the book is 8.2 m/s. Where am I going wrong please.

When you substituted for $\cos\alpha$ towards the end, you used 6/10, instead of 8/10.

Edit: PS. Just because the ring can move does not imply the tensions are the same. It is the fact that the ring is smooth (i.e. no friction) that the tensions are the same.