The product format A2 - B2 is a special form of algebraic expression.
It is equal to (A + B)(A - B).
Many other expressions, like the cos2A = cos2A - sin2B,
can also be expressed in the special product form, that is,
cos2A = (cos A + sin A)(cos A - sin A).
But, how about A2 + B2 ?
Can it be expressed in the (A + B)(A - B) format?
However we need to deviate a bit from the norm.
We need to know the imaginary "i" in complex number system.
Click here for a review to it.
Since i2 = -1, we can make use of this property for the special product A2 + B2.
Here it goes....
A2 + B2 = A2 - (i2)(B2)
Using the basic of Indices, it can be modified to,
This becomes, therefore, (A + iB)(A - iB).
A2 - B2 = (A + B)(A - B), and
A2 + B2 = (A + iB)(A - iB).
The principles of the special product still holds regardless of the addition or subtraction operation between A square and B square.
Student: What an interesting twist is mathematics in this matter. Anything seems to be simple if we know the technique!