Problem 75 - Precalculus Essentials

Question

Explain how to verify an identity.

Step-by-Step Solution

Answer

Choosing the more complex side to simplify first, using established algebraic operations and identities to manipulate the chosen side until it matches the other side, is the process of verifying an identity. The process can require working on both sides of the equation depending on the complexity.

Step 1: Understand the equation

Identify the given two mathematical expressions that need to be proved identical.

Step 2: Choose a side

Choose the more complex side of the equation to begin manipulating first. This is easier than trying to make the simpler side more complex.

Step 3: Apply mathematical operations and identities

Use the established algebraic laws such as the distributive law, associative law, commutative law, and identities like sin^2x + cos^2x = 1, etc., to simplify the chosen side of the equation.

Step 4: Simplification

Keep simplifying the expressions by performing the operations such as addition, subtraction, multiplication, division, factoring, expanding etc., and using identities until it reaches a form that can be compared with the other side.

Step 5: Compare the two sides

After simplifying, compare the manipulated side with the other side. If they are identical, then the initial identity is verified and confirmed.

Step 6: If necessary, manipulate the other side

In the case where the sides do not match after simplifying, try simplifying or manipulating the other side of the equation until it matches the already manipulated one. If they match, the identity is confirmed; if not, the original equation may not be an identity.