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.

Another way would be this
\begin{aligned}
(\sin p + \cos p)^{10} &= (\sin p + \cos p)^{2\times5}\\
&=(\sin^2p+\cos^2p+2\sin p\cos p)^5\\
&=(1+\sin{2p})^5
\end{aligned}
Since $-1\leq \sin{2p}\leq 1$ then $$0\leq(1+\sin{2p})^5=(\sin p + \cos p)^{10}\leq32$$ where $p=-\pi/4$ and $p=\pi/4$ correspond to the minimum and maximum, respectively, though not unique. 
