Let $P(n)=A^{\mathbb{n}}=\left[\begin{array}{cc}\cos n \theta & \sin n \theta \\ -\sin n \theta & \cos n \theta\end{array}\right]$, for all $n \in N$.
Step 1: Put $n=1$
$\therefore \quad$ R.H.S. $=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & \cos \theta\end{array}\right]=\mathbf{A}=$ L.H.S.
$\therefore \quad P(n)$ is true for $n=1$.
Step 2: Let us consider that $\mathrm{P}(\mathrm{n})$ is true for $\mathrm{n}=\mathrm{k}$.
$\therefore \quad A^k=\left[\begin{array}{cc}\cos k \theta & \sin k \theta \\ -\sin k \theta & \cos k \theta\end{array}\right]$
...(i)
Step 3: We have to prove that $P(n)$ is true for
$\mathrm{n}=\mathrm{k}+\mathrm{l}$
i.e., to prove that
$\begin{aligned} & A^{k+1}=\left[\begin{array}{cc}\cos (k+1) \theta & \sin (k+1) \theta \\ -\sin (k+1) \theta & \cos (k+1) \theta\end{array}\right] \\ & \text { R.H.S. }=\left[\begin{array}{cc}\cos (k+1) \theta & \sin (k+1) \theta \\ -\sin (k+1) \theta & \cos (k+1) \theta\end{array}\right] \\ & \text { L.H.S. }=A^{k+1}=A^k \cdot A\end{aligned}$
$\begin{aligned} & =\left[\begin{array}{cc}\cos k \theta & \sin k \theta \\ -\sin k \theta & \cos k \theta\end{array}\right]\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right] \quad \ldots[\text { From (i)] } \\ & =\left[\begin{array}{cc}\cos k \theta \cos \theta-\sin k \theta \sin \theta & \cos k \theta \sin \theta+\sin k \theta \cos \theta \\ -\sin k \theta \cos \theta-\cos k \theta \sin \theta & -\sin k \theta \sin \theta+\cos k \theta \cos \theta\end{array}\right] \\ & =\left[\begin{array}{cc}\cos k \theta \cos \theta-\sin k \theta \sin \theta & \sin k \theta \cos \theta+\cos k \theta \sin \theta \\ -(\sin k \theta \cos \theta+\cos k \theta \sin \theta) & \cos k \theta \cos \theta-\sin k \theta \sin \theta\end{array}\right]\end{aligned}$
$\begin{aligned} & =\left[\begin{array}{cc}\cos (k \theta+\theta) & \sin (k \theta+\theta) \\ -\sin (k \theta+\theta) & \cos (k \theta+\theta)\end{array}\right] \\ & =\left[\begin{array}{cc}\cos (k+1) \theta & \sin (k+1) \theta \\ -\sin (k+1) \theta & \cos (k+1) \theta\end{array}\right]\end{aligned}$
$=$ R.H.S.
$\therefore \quad P(n)$ is true for $n=k+1$.
Step 4: From all steps above, by the principle of
Mathematical induction, $P(n)$ is true for all
$\mathrm{n} \in \mathrm{N}$.
$\therefore \quad A^n=\left[\begin{array}{cc}\cos n \theta & \sin n \theta \\ -\sin n \theta & \cos n \theta\end{array}\right]$ for all $n \in N$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.