In this lesson you learned about mathematical proofs, which are at the core of mathematics. A proof is just an argument showing that a statement must be true. Once you know the statement is always true, you can use the statement in future scenarios, such as simplifying expressions and solving other equations.
For the type of proof you learned in this lesson, the goal is to show that both sides of the identity are equal by manipulating each side independently until they are the same expression. There is no specific pattern for how to proceed with a given proof. A bit of luck, determination, and some intuition will help you complete a proof.
In the next module you will begin to explore exponents and their inverse, logarithms.