Uniform convergence help

Question

Is this true ? please tell me how they got here, this is to prove the uniform convergence of the function (point wise limit is the function zero):

$$\text{for }x \in [0,\pi], \sup\left|\sin(x)^n\left(1-\sin(x)^n\right)\right|=\frac1{4^n}$$

Answer 1

No, it is not true. Take $x=\arcsin\left(\sqrt[n]{\frac12}\right)$. Then$$\sin^n(x)\bigl(1-\sin^n(x)\bigr)=\frac14.$$