To find $p$ such that max/min of $(\sin p+\cos p)^{10}$ occurs To find max/min of $(\sin p+\cos p)^{10}$. I have to find value of $p$ such that the expression is max/min. I tried to manipulate expression so as to get rid of at least $\sin$ or $\cos$. Then I can put what is left over equals to $1$ to get the maximum. But I'm unable to do that.

$(\sin p + \cos p)^{10} = (\sin^2 p + 2\sin p\cos p + \cos^2 p)^5 = (1+\sin 2p)^5$
Function $x\mapsto (1+x)^5$ is monotone increasing, thus, extremes of $(1+\sin 2p)^5$ are the same as extremes of $\sin 2p$.
