How do one prove the following trigonometric identity with the standard rotation matrix $T_{θ}$
$cos(2θ) = cos^{2}(θ) − sin^{2}(θ)$?
The hint given is to compare $T_{2θ}$ and $T_{θ}◦T_{θ}$. We have not learned about complex numbers yet.
I have come to that:
$T_{θ}◦T_{θ}$= $\begin{bmatrix}cos^{2}\theta -sin^{2}\theta & -2cos{}\theta sin \theta \\2sin \theta cos{}\theta & cos^{2}\theta -sin^{2}\theta \end{bmatrix}$
$T_{2θ}$= $\begin{bmatrix}cos{2}\theta & -sin{2}\theta \\sin{2}\theta & cos{2}\theta \end{bmatrix}$
Is it enough to show this and say that one element in the matrix equals the other element in the other matrix?
Your mistake is in computing $T_\theta\circ T_\theta$. Specifically, the top left and bottom-right entries are NOT 0.