"Problems3-Question1g
In solution, it has only the set of sin(t) and sin(2t) that is a basis.
Why it doesn’t have the set of sin(t) and sin(t)cos(t)?
Why the set of sin(t) and sin(t)cos(t) is not a basis?"

Both are fine. You can have more than one basis for a given vector space.

I chose to write {sin t, sin 2t, sin t cos t} = {sin t, sin 2t, (1/2)sin 2t } >>> Can see that a basis is {sin t, sin 2t}.

It is equally valid to say {sin t, sin 2t, sin t cos t} = {sin t, 2sin t cos t, sin t cos t } >>> Can see that a basis is {sin t, sin t cos t}