Question

Let $X$ be the solution set of the equation $A^x=I$, where $A=\left[\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]$ and $I$ is the corresponding unit matrix and $x\in N$ then find the minimum value of $\sum ({\cos }^x\theta +{\sin }^x\theta ),\theta \in R$.

Answer 1

$A^2=I\Rightarrow x=2,4,6,.......$

$\sum ({\cos }^x\theta +{\sin }^x\theta )$

$=({\cos }^2\theta +{\sin }^2\theta )+({\cos }^4\theta +{\sin }^4\theta )+.......$

$=({\cot }^2\theta +{\tan }^2\theta )\ge 2$