How to Simplify Trigonometric Expressions Using Identities

Trigonometric simplification is one of the most important skills in Pre-Calculus. At first, these questions can look intimidating because they often contain squared trig functions, double angles, fractions, or products of sine and cosine.

However, most of these questions are based on recognizing a small set of common patterns. The key is not to randomly expand everything. Instead, look for familiar identity structures.

In many questions, your goal is to rewrite the expression as a single trigonometric function, such as:

sin(60°), cos(2x), and tan(4θ)

or as a single primary trigonometric function, meaning the final answer should involve only one of:

sin(x), cos(x), and tan(x)

Key Identities to Know

The most common identities used in these simplification questions are the double-angle identities:

sin(2x) = 2sin(x)cos(x)

cos(2x) = cos²(x) − sin²(x)

cos(2x) = 2cos²(x) − 1

cos(2x) = 1 − 2sin²(x)

tan(2x) = (2tan(x)) / (1 − tan²(x))

The sum and difference identities are also useful:

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

sin(x − y) = sin(x)cos(y) − cos(x)sin(y)

cos(x + y) = cos(x)cos(y) − sin(x)sin(y)

cos(x − y) = cos(x)cos(y) + sin(x)sin(y)

tan(x + y) = (tan(x) + tan(y)) / (1 − tan(x)tan(y))

tan(x − y) = (tan(x) − tan(y)) / (1 + tan(x)tan(y))

The strategy is simple: look for a pattern, identify the matching identity, then replace the expression with the simpler version.

Example 1: Recognizing the Sine Double-Angle Identity

Simplify:

2sin(35°)cos(35°)

This matches the identity:

sin(2x) = 2sin(x)cos(x)

In this case:

x = 35°

So:

2sin(35°)cos(35°) = sin(2 ⋅ 35°) = sin(70°)

Final answer:

sin(70°)

The important clue was the structure:

2sin(x)cos(x)

Whenever you see 2 times sine of an angle times cosine of the same angle, you should immediately think of the sine double-angle identity.

Example 2: Simplifying to a Single Primary Trig Function

Simplify:

(1 + cos(2x)) / (2cos(x))

At first, this looks like a fraction. However, the numerator contains:

1 + cos(2x)

We know that:

cos(2x) = 2cos²(x) − 1

Rearranging that identity gives:

1 + cos(2x) = 2cos²(x)

Now substitute this into the original expression:

(1 + cos(2x)) / (2cos(x)) = (2cos²(x)) / (2cos(x))

Cancel the common factor:

(2cos²(x)) / (2cos(x)) = cos(x)

Final answer:

cos(x)

This example shows why it is helpful to know multiple forms of the cosine double-angle identity. Sometimes the expression will not appear exactly as the identity is written. You may need to recognize a rearranged version.

Example 3: Using the Cosine Subtraction Identity

Simplify:

cos(7θ)cos(3θ) + sin(7θ)sin(3θ)

This expression matches the identity:

cos(x − y) = cos(x)cos(y) + sin(x)sin(y)

Here:

x = 7θ

and

y = 3θ

So:

cos(7θ)cos(3θ) + sin(7θ)sin(3θ) = cos(7θ − 3θ)

Now simplify inside the brackets:

cos(7θ − 3θ) = cos(4θ)

Final answer:

cos(4θ)

The key pattern was:

cos(x)cos(y) + sin(x)sin(y)

This becomes:

cos(x)

Final Strategy

When simplifying trigonometric expressions, ask yourself these questions:

1. Do I see 2sin(x)cos(x)?

   Use: sin(2x) = 2sin(x)cos(x)

2. Do I see cos²(x) − sin²(x), 2cos²(x) − 1, or 1 − 2sin²(x)?

   Use a cosine double-angle identity.

3. Do I see (2tan(x)) / (1 − tan²(x))?

   Use: tan(2x) = (2tan(x)) / (1 − tan²(x))

4. Do I see something like cos(x)cos(y) + sin(x)sin(y)?

   Use the corresponding sum or difference identity.

The main idea is that trig simplification is pattern recognition. Once you know which pattern you are looking at, the expression often simplifies in only one or two steps.

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