In lecture my professor wrote down this equation and it's solution: $$A_x^{'}=\sqrt{A_x^2+A_y^2}\cos(\theta +\phi)=A_x^{'}=\sqrt{A_x^2+A_y^2}\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)$$
$$=A_x\cos(\phi)-A_y\sin(\phi)$$
All of the steps are skipped in both lecture and in the textbook. That's the only information given. I've been trying to work out the algebra but it's not happening. I tried squaring both sides, it got very messy, but no matter what I tried I couldn't get it down to the solution given.
I tried using the product to sum of two angles trig identity, but it kept me going in circles. What is the best way to get started on simplifying this?
Note that $\frac{A_{x}}{\sqrt{(A_{x})^{2}+(A_{y})^{2}} }\leq1 $$ $ So, if $\cos(\theta)$ is set to $\frac{A_{x}}{\sqrt{(A_{x})^{2}+(A_{y})^{2}} }$, then it's not hard to see that $\sin(\theta)=\frac{A_{y}}{\sqrt{(A_{x})^{2}+(A_{y})^{2}}}$. Thus,$\sqrt{(A_{x})^{2}+(A_{y})^{2}}(\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi))$ reduces to $A_{x}\cos(\phi)-A_{y}\sin(\phi)$.