10.4 Trigonometric Identities

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  • Understand what a trigonometric identity is and how it differs from a regular equation
  • Know and use the three key Pythagorean identities:
    • sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1
    • sec⁡2A=1+tan⁡2A\sec^2 A = 1 + \tan^2 A
    • cosec⁡2A=1+cot⁡2A\cosec^2 A = 1 + \cot^2 A
  • Rearrange each identity into alternative useful forms
  • Use these identities to prove more complex trigonometric statements
  • Use these identities to simplify trigonometric expressions

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1. What Is a Trigonometric Identity?

A trigonometric identity is a mathematical equation involving trigonometric functions (like sine, cosine, or tangent) that is always true, no matter what value you substitute for the angle.

Think of it this way:

  • A regular equation like 2x=102x = 10 is only true for one value (x=5x = 5).
  • An identity like sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1 is true for every value of xx.

This "always true" property is what makes identities so powerful — you can use them to rewrite or simplify expressions at any point in a problem.

📌 Important: When you are asked to prove an identity, you are showing that the left-hand side (LHS) and the right-hand side (RHS) are equal by manipulating one side until it looks exactly like the other.

2. Introducing the Reciprocal Trigonometric Functions

Before you can fully understand the three identities, you need to meet three new trigonometric functions. These are called reciprocal functions — meaning they are simply "1 divided by" a familiar function.

FunctionHow It's DefinedRead As
sec⁡A\sec Asec⁡A=1cos⁡A\sec A = \dfrac{1}{\cos A}"sec A"
cosec⁡A\cosec Acosec⁡A=1sin⁡A\cosec A = \dfrac{1}{\sin A}"cosec A"
cot⁡A\cot Acot⁡A=cos⁡Asin⁡A\cot A = \dfrac{\cos A}{\sin A}"cot A"

Let's understand each one clearly:

Secant (sec⁡A\sec A): This is simply 1 divided by cos⁡A\cos A. For example, if cos⁡A=0.5\cos A = 0.5, then sec⁡A=10.5=2\sec A = \frac{1}{0.5} = 2.

Cosecant (cosec⁡A\cosec A): This is 1 divided by sin⁡A\sin A. For example, if sin⁡A=0.25\sin A = 0.25, then cosec⁡A=10.25=4\cosec A = \frac{1}{0.25} = 4.

Cotangent (cot⁡A\cot A): This is cos⁡A\cos A divided by sin⁡A\sin A. You can also think of it as cot⁡A=1tan⁡A\cot A = \frac{1}{\tan A}, since tan⁡A=sin⁡Acos⁡A\tan A = \frac{\sin A}{\cos A}.

⚠️ Watch out: sec⁡A\sec A is NOT the same as sin⁡A\sin A. Similarly, cosec⁡A\cosec A is NOT the same as cos⁡A\cos A. The names look similar but they mean very different things!

Also remember that: tan⁡A=sin⁡Acos⁡A\tan A = \frac{\sin A}{\cos A}

This definition of tangent is itself a useful identity and will appear frequently in proofs.

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