It seems to me as if you're making a mistake I see all too often, in thinking that sin(4t) represents multiplication, such that it would be \(\displaystyle sin \cdot 4t\), but that's not the case. Sine is a function, and 4t is its argument. You're squaring the function, not the argument. You can, of course, see that your proposed inequality doesn't work, by setting \(\displaystyle t=1\):
\(\displaystyle sin(4t)=sin(4) \approx -0.7568\)
\(\displaystyle sin^2(4t)=sin(4) \cdot sin(4) \approx (-0.7568)^2 \approx 0.5728\)
\(\displaystyle sin^2(16t)=sin(16) \cdot sin(16) \approx (-0.2879)^2 \approx 0.0829\)
As it turns out, the equality does hold true for many values of t, but that's only due to the periodic nature of the sine function, and it doesn't hold true for all values of t. Another thing you might play around with is squaring other functions, to confirm the idea that the argument remains unchanged when you square a function. Let \(\displaystyle f(x)=x^2\):
\(\displaystyle f(2x)=(2x)^2=4x^2\)
\(\displaystyle f^2(2x)=f(2x) \cdot f(2x) = (2x)^2 \cdot (2x)^2=16x^4\)
\(\displaystyle f^2(4x^2)=f(4x^2) \cdot f(4x^2) = (4x^2)^2 \cdot (4x^2)^2=256x^8\)
