Geometric Proof of $a^2-b^2=(a+b)(a-b )$ and its applications I am teaching a student on the subject of factoring. One commonly used formula is $a^2-b^2=(a+b)(a-b) $. It is easy to prove it from RHS to LHS algebraically, but how to prove it geometrically? I would also like to find some of its applications, such as this:$(\sin\theta)^2+(\cos\theta)^2=( \cos \theta +i\sin \theta)(\cos\theta -i\sin \theta )=e^{i\theta}\cdot e^{-i\theta  } =e^0=1$
Are there any other good examples?
EDIT: some comments for applications
(Albus) $\frac{1}{\sqrt a+b}=\frac{\sqrt a-b}{a-b^2}$
(Paul) $a\times b=(\frac{a+b}{2})^2-(\frac{a-b}{2})^2$
(other ) $x=x’+y’, y=x’-y’, xy=x’^2-y’^2=1$ Hyperbola  equation
 A: Hint:
$$a^2 - b^2 =  (a-b)b + (a-b)b + (a-b)^2 $$
$$ = (a-b)[b + b + (a-b)] $$
or
$$a^2 - b^2 =  (a-b)a + (a-b)b$$
$$ = (a-b)(a+ b) $$
Draw two squares inside each other ($a>b$) and look at the three or two leftover rectangular areas.
A: 
Here is one example I can think of. Ask the student to calculate the area of the yellow zone in the above figure: the area between two squares with sides $a$ and $b$ . It must be easy for the student to obtain the result by subtracting the areas of the two squares:
$$S_{yellowZone} = a^2 - b^2 \qquad (1)$$
But we can also calculate the area as the sum of areas of two narrow rectangles $EBHF$ and $GHCD$ :
$$S_{EBHF} = EB \times EF = (a-b).b$$
$$S_{GHCD} = GD \times GH = (a-b).a$$
Adding up these two we calculate the yellow zone area, which we had previously found to be $a^2 - b^2$:
$$S_{yellowZone} = (a-b).b + (a-b).a = (a-b)(a+b) \qquad (2)$$
From (1) and (2) we have our proof.
Edit: I just noticed that user973676 suggested this approach too. Credits to them.
So here is another example! Calculate the area of the rectangle $AXYZ$ in the figure below:

Update: One useful application of this identity, in addition to being handy in arithmetic calculations, is that it helps teach students that if the sum of two numbers is constant, then their product is maximum when the two numbers are equal.
It also opens a door for teaching the AM-GM inequality.
