1. The problem is to understand the concept of an identity in algebra. 2. An identity is an equation that is true for all values of the variable involved. 3. For example, the distributive property is an identity: $$a(b+c) = ab + ac$$ for all values of $a$, $b$, and $c$. 4. Another example is the Pythagorean identity in trigonometry: $$\sin^2(x) + \cos^2(x) = 1$$ for all $x$. 5. Identities are useful because they allow us to simplify expressions and solve equations more easily. 6. To verify an identity, you can start with one side of the equation and use algebraic manipulations to transform it into the other side. 7. For instance, to verify $$\sin^2(x) + \cos^2(x) = 1$$, recall the definition of sine and cosine on the unit circle, which confirms the identity. 8. Understanding identities helps in recognizing equivalent expressions and solving problems efficiently.