Hi,

there are some parenthesis that need to be changed and Sqrt is capitalised.

Solve[(1 - t)/(Sqrt[1 + t^2])*(Sqrt[2]) == 1/2, t]

gives

{{t -> 1/7 (8 - Sqrt[15])}}

or

{{t -> 0.589574}}

which is different from your solution.

In fact, your suggested solution does not appear to solve the equation:

In[90]:= (1 - t)/(Sqrt[1 + t^2])*(Sqrt[2]) == 1/2 /. t -> 2 - Sqrt[3]

Out[90]= False

In[93]:= (1 - t)/(Sqrt[1 + t^2])*(Sqrt[2]) == 1/2 /.

t -> 1/7 (8 - Sqrt[15]) // N

Out[93]= True

M.