Reciprocal & Quotient Identities

Rewrite trig expressions using identities and simplification. Math 30-1 Alberta.

Lesson 5.1 of Trigonometric Identities in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• What is a Trigonometric Identity?. Learn what makes an equation an identity, understand the difference between numerical verification and algebraic proof, and discover the golden rule for proving identities.
• Reciprocal & Quotient Identities. Explore the fundamental reciprocal identities (csc, sec, cot) and quotient identities (tan, cot) that form the foundation of trigonometric manipulation.
• Simplifying Expressions & Non-Permissible Values. Master algebraic strategies for simplifying trig expressions and learn how to identify non-permissible values (NPVs) that make denominators equal to zero.
• Simplifying to a Single Ratio or Value. Push simplification all the way — reduce complex trig expressions down to a single ratio or numerical value using reciprocal identities, Pythagorean identities, and algebraic strategies.

Inside the lesson — a free preview

What is a Trigonometric Identity?

Definition. An identity is an equation containing one or more variables that is true for ALL values for which both sides of the equation are defined.

The Core Concept. Left Side = Right Side

No matter what values you substitute (as long as they're defined), both sides will always be equal.

Three Ways to Verify an Identity

There are three ways to check if an equation might be an identity — but only one actually proves it.

1️⃣ Numerical Verification. Numerical Verification means substituting a specific value into both sides to check if they are equal. If both sides match for that value, it suggests the identity might be true.

Example of Numerical Verification. Checking that (a + b)^2 = a^2 + 2ab + b^2 using:

a = 5 and b = 2

Left Side:

(5 + 2)^2 = 7^2 = 49

Right Side:

5^2 + 2(5)(2) + 2^2 = 25 + 20 + 4 = 49

Both sides equal 49 ✓

⚠️ The Problem. This only proves it works for a = 5 and b = 2 — it does NOT prove it's an identity.

To be an identity, it must work for ALL values, not just one example.

2️⃣ Graphical Verification. Graphical Verification means graphing the function from each side of the identity. If the two graphs perfectly overlap across the domain, the identity appears to be true.

However, just like numerical verification, this still does not prove the identity is true for all values — graphs can only show a limited window.

3️⃣ Algebraic Proof. Algebraic Proof is the only method that truly proves an identity is true for all values where the variable is defined.

You use algebraic operations to manipulate one side until it looks identical to the other side.

Worked example: Example

Prove that (a + b)² = a² + 2ab + b². We'll work on the left side only and manipulate it until it matches the right side.

Conclusion. LS = RS ✓

Since we algebraically manipulated only the left side until it matched the right, this is proven to be an identity for all real values of a and b.

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