Problem 75 - Algebra and Trigonometry

Question

Explain how to verify an identity.

Step-by-Step Solution

Answer

To verify an identity, choose either the more complex side or any side if both are equally complex, simplify the side using known identities and basic operations, and then compare with the other side to confirm if the identity is verified.

Step 1: Understand the Problem

An identity is an equation that is true for all values of the variable in its domain. Verify means to prove or check whether it is true. However, remember that you can perform operations on both sides independently to make them look identical.

Step 2: Start On One Side

Choose the more complicated side of the equation. If both sides are equally complex, choose either side. Make this side, your 'working-side' from now on.

Step 3: Use Known Identities

Use the appropriate identities in order to simplify the working side of the equation. You may replace sin^2 x with 1 - cos^2 x or cos^2 x with 1 - sin^2 x. Similarly if you have a tan x = sin x/cos x, sin^2 x + cos^2 x = 1 and many more.

Step 4: Simplify The Equation

Simplify the working side as much as possible using responses from elementary operations such as addition, subtraction, multiplication and division. Break up fractions or combine fractions as needed.

Step 5: Compare With Other Side

After simplification, the working side should be the same as the non-working side. If both the sides are not identical, check the work. If they look identical, the identity is verified.