How to Factor and Simplify Trigonometric Expressions

A Step-by-step Guide to Factoring and Simplifying Trigonometric Expressions

Below is a step-by-step guide to help you understand how to do this:

Step 1: Recognize the Type of Expression

The first step is to identify what kind of trigonometric expression you’re dealing with. The expression could involve \(sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (csc)\), or \(cotangent (cot)\). Also, look out for quadratic expressions (e.g. \(sin^2x + cos^2x\)), product-to-sum forms, and double- or half-angle forms.

Step 2: Identify Appropriate Trigonometric Identities

To factor and simplify these expressions, you need to understand and apply trigonometric identities. There are several main categories of these:

• Pythagorean identities:
  1. \(sin^2x + cos^2x = 1\)
  2. \(tan^2x + 1 = sec^2x\)
  3. \(1 + cot^2x = csc^2x\)
• Reciprocal identities:
  1. \(cscx=\frac{1}{sinx}\)
  2. \(secx=\frac{1}{cosx}\)
  3. \(cotx=\frac{1}{tanx}\)
• Quotient identities:
  1. \(tanx=\frac{sinx}{cosx}\)
  2. \(cotx=\frac{cosx}{sinx}\)
• Co-Function identities:
  1. \(sin(90^°-x)=cosx\)
  2. \(cos(90^°-x)=sinx\)
• Even-Odd identities:
  1. \(sin(-x) = -sinx\)
  2. \(cos(-x) = cosx\)
  3. \(tan(-x) = -tanx\)
• Double-Angle identities:
  1. \(sin2x = 2sinxcosx\)
  2. \(cos2x = cos^2x – sin^2x = 2cos^2x – 1 = 1 – 2sin^2x\)
  3. \(tan2x =\frac{2tanx}{(1 – tan^2x)}\)
• Half-Angle identities:
  1. \(\frac{sinx}{2}=±\sqrt(\frac{1-cosx}{2})\)
  2. \(\frac{cosx}{2}=±\sqrt(\frac{1+cosx}{2})\)
  3. \(\frac{tanx}{2}=±\sqrt(\frac{1-cosx}{1+cosx})=\frac{1-cosx}{sinx}=\frac{sinx}{1+cosx}\)
• Product-to-Sm and Sum-to-Product identities.

Step 3: Apply the Appropriate Identity

Once you’ve identified which identity applies, you substitute it into the expression. For example, if you have an expression like \(sin²x + cos²x\), you can directly replace it with \(1\) according to the Pythagorean identity.

Step 4: Simplify the Expression

After substitution, simplify the resulting expression if possible. You might combine like terms, use algebraic factoring methods, or apply additional identities.

Step 5: Check Your Work

Substitute some values for the variable in the original and the simplified expression to check if they are indeed equal. If they are, then you’ve correctly simplified the trigonometric expression.

Example

Let’s walk through an example:

We’re given the expression \(sin²x – 2sinx + 1\).

Step 1: Identify the expression. This looks like a quadratic expression in sinx\).

Step 2: Recall an identity. This is similar to the square of a binomial, \((a – b)^2= a^2- 2ab + b^2\). Here, \(a = sinx\) and \(b = 1\).

Step 3: Apply the identity. We can rewrite the given expression as \((sinx – 1)^2\).

That’s it! We’ve factored and simplified the given trigonometric expression.

By developing a strong foundation in trigonometric identities and practicing factoring and simplification, you’ll be able to tackle a wide range of trigonometric expressions.