Quaternions concatenation is the sum of angles ? (2)

Hello,

With the two quaterions q1=q2=(cos theta/2, sin theta/2, 0, 0)
Does the product q1q2 yield a third quaternion q3 equals (cos theta, sin theta, 0, 0) ?

This is the result I am expecting to have from my quaternion class, but I don't have it.
My expectation is based on the fact that the concatenation of q1 and q2 will yield a quaternion representing both rotations. That means, for me, the sum of the angles each quaternion represents if the rotations are around the same vector.

So, does the product q1q2 yield a third quaternion q3 equal (cos theta, sin theta, 0, 0) ?
Or is there anything wrong with my assumption ?

Thank you in advance