For example, $\sin(x)\cos(x)$ can be written as $\sin(2x)/2$, the limit as $x$ approaches $0$ of $\sin(x)\cos(x)$ is $0$, and the limit as x approaches $\pi/2$ is $0$. I don't see a reason why sine always increases/decreases faster than cosine. Why does sine overpower cosine near x=0, and why does cosine overpower sine near x=pi/2, I feel that if they did increase the same the limit should be somewhere in the middle, such as 1/2 for both of these values. If you try to take the derivative of each with respect to some other to see why one increases faster, you end up with the same problem because the derivatives just keep giving you negative versions of the original problem.
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the limit of $\sin(x)\cdot \cos(x)$ as $x\rightarrow \pi /2$ is $0$. So in both cases you have $\sin(x)\cdot \cos(x)=0$. Infact they do increase/decrease with the same rate. Infact $\sin(\pi/2-x)=\cos(x)$ if you vary $x$ they have the same values so they increase/decrease with the same rate.