# If $\cos a = \cos a \cos b - \sin a \sin b$, then $b = 2n\pi$

The solutions to a tutorial question I am working on are as follows:

$$\cos\left(\frac{p}{\sqrt{\eta}}x\right) = \cos\left(\frac{p}{\sqrt{\eta}}x\right)\cos\left(\frac{p}{\sqrt{\eta}}2L\right)-\sin\left(\frac{p}{\sqrt{\eta}}x\right)\sin\left(\frac{p}{\sqrt{\eta}}2L\right)$$

Clearly if LHS is equal to RHS, then

$$\frac{2Lp}{\sqrt{\eta}} = 2n \pi$$

I am confused as to how this is clear. If anyone is able to provide more working that would be great.

Note: (From comment). The question was originally about concluding $\cos\left(\frac p\eta x\right)$, given $$\cos\left(\frac p\eta x\right)= \cos\left(\frac p{\eta} x +\frac p{\eta}2L\right)$$ The solution then expanded by using the sum of angles formula for $\cos$.

• Simply apply identity $\cos a \cdot \cos b - \sin a \cdot \sin b = \cos(a+b)$, where $a=\dfrac{p}{\sqrt{\eta}}x$ and $b=\dfrac{p}{\sqrt{\eta}} 2L$. – Oleg567 May 15 '18 at 16:52
• @Oleg567 the solutions started in that form though i.e. with cos(a +b) and expanded it out – SFL May 15 '18 at 16:55
• There are two answers you wrote only one but its correct – Abhishek May 15 '18 at 16:57
• @Blue Nice, succinct, on point, title! – Namaste May 15 '18 at 18:08

Hint: The RHS gives $$\cos\left(\frac p{\eta} x +\frac p{\eta}2L\right)^{(\dagger)}$$

So what can you conclude if $$\cos\left(\frac p\eta x\right) = \cos\left(\frac p{\eta} x +\color{blue}{\frac p{\eta}2L}\right)\;?$$

This equality only occurs when $\color{blue}{\dfrac {2Lp}{\eta}} = 2n\pi$, where $n$ is an integer. (Note that $2\pi (n)$ is $n$ full revolutions of the unit circle).

$^{(\dagger)}$ If this is how the problem originally appeared, I'd suggest NOT expanding using the sum of angles identity for $\cos$.)

• E.g., What is $\cos\left(\frac \pi 4 + 2\cdot 2\pi\right)$? It is $\cos \left(\frac \pi 4\right) = \frac{\sqrt 2}2$. What is $\cos\left(\frac \pi 3 -3\cdot 2\pi\right)?$ It is $\cos\left(\frac \pi 3\right) = \frac 12$. – Namaste May 15 '18 at 17:12
• Another way to think of this, apart from cos, is what time is it tomorrow, 24 hours from now (one full revolution of the hands on the clock)? (The hands on the clock will be in the exact same position they are now, 24 hours from now. In the unit circle, any angle $0 \leq \theta\lt 2\pi$ $\pm$, plus or minust (plus any number of full revolutions, counter clockwise or clockwise) will result in the same angle, and so the cos of the initial angle will equal the cos of the final angle after $n$ full revolutions (after $\pm n 2\pi$). – Namaste May 15 '18 at 17:34

Let $\frac{p}{\sqrt{\eta}}=l$
Then, $RHS=\cos(lx+2Ll)$
$\therefore \cos(lx+2Ll)=\cos(lx)$
$\therefore lx+2Ll=2n\pi \pm lx$
$\therefore 2Ll=2\eta\pi,2\eta\pi-2lx$
$\frac{2Lp}{\sqrt{\eta}}=2\eta\pi,2\eta\pi-2\frac{p}{\sqrt{\eta}}x$