<s>[INST] Trigonometric inequalities in a triangle What is the proof of
$~~\cos^2 A+\cos^2 B+\cos^2 C \leq 1 ~~$
in an acute triangle ?
This will be of help in finding the answer (if such exists) to finding the minimal T in any ∆ ABC when
$$T \geq \sin^k  A+ \sin^k  B+ \sin^k  C~ ,~~~~ k \geq 3$$
 [/INST] We have
\begin{align}
\cos^2A+\cos^2B+\cos^2C&=\frac12(1+\cos2A)+\frac12(1+\cos2B)+\cos^2C\\
&=\frac12(2+\cos2A+\cos2B)+\cos^2C\\
&=\frac12(2+2\cos(A-B)\cos(A+B))+\cos^2C\\
&=1+\cos(A-B)\cos(\pi-C)+\cos^2C\\
&=1-\cos(A-B)\cos C+\cos^2C\\
&=1-\cos C(\cos(A-B)-\cos(\pi-(A+B)))\\
&=1-\cos C(\cos(A-B)+\cos(A+B))\\
\cos^2A+\cos^2B+\cos^2C&=1-2\cos A\cos B\cos C\\
\implies\cos^2A+\cos^2B+\cos^2C&\leq1\\
\end{align}
</s>