Perform the indicated operation and simplify the result so that there are no quotients. \[ (5+\sin t)^{2}+\cos ^{2} t \] The simplified form, with no quotients, of $(5+\sin t)^{2}+\cos ^{2} t$ is (Do not factor.)

Step 1 ：Given the expression \((5+\sin t)^{2}+\cos ^{2} t\).

Step 2 ：Expand the square to get \(\sin(t)^2 + 10\sin(t) + \cos(t)^2 + 25\).

Step 3 ：Use the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\) to simplify the expression further.

Step 4 ：The simplified form, with no quotients, of \((5+\sin t)^{2}+\cos ^{2} t\) is \(\boxed{10\sin t + 26}\).