TRIGONOMETRIC IDENTITIES

Identities and Equations

An equation such as y2 – 9y + 20 = (y – 4)(y – 5) is an identity because the left-hand side (LHS) is equal to the right-hand side (RHS) for whatever value is substituted to the variable.

Based on the example, an identity is defined as an equation, which is true for all values in the domain of the variable.

Identities and Equations

There are identities which involve trigonometric functions. These identities are called trigonometric identities.

Trigonometric identity is an equation that involves trigonometric functions, which is true for all the values of θ for which the functions are defined.

Identities and Equations

A conditional equation is an equation that is true only for certain values of the variable.

The equations y2 – 5y + 6 = 0 and x2 – x – 6 = 0 are both conditional equations. The first equation is true only if y = 2 and y = 3 and the second equation is true only if x = 3 and x = -2.

The Fundamental Identities

The Fundamental Identities

Reciprocal Identities

Reciprocal Identities Equivalent Forms Domain Restrictions

Quotient (or Ratio) Identities

Quotient Identities Domain Restrictions

Pythagorean Identities

Negative Arguments Identities

Notes:

The real number x or θ in these identities may be changed by other angles such as α, β, γ, A, B, C,….

The resulting identities may then be called trigonometric identities.

Example:

Find the remaining circular functions of θ using the fundamental identities, given sin θ = and P(θ) ϵ II.

Simplifying Expressions

Examples:

Simplify the following expressions using the fundamental identities.

1. tan3 x csc3 x

2. sec x • cos x – cos2 x

3. (csc2 x – 1)(sec2 x sin2 x)

4.

5. (cos θ – 1)(cos θ + 1)

6. sin2 θ + cot2 θ sin2 θ

Proving Identities There is no exact procedure to be followed in

proving identities. However, it may be helpful to express all the given functions in terms of sines and cosines and then simplify.

To establish an identity, we may use one of the following:

1. Transform the left member into the exact form of the right.

2. Transform the right into the exact form of the left, or

3. Transform each side separately into the same form.

Examples

1. Prove that + = is an identity.

2. Verify if tan2 β – sin2 β = tan2 β sin2 β is an identity.

Exercises

1. If sin θ = and P(θ) is in quadrant IV, find the other trigonometric function values of θ using the fundamental identities.

2. Express cos θ (tan θ – sec θ) in terms of sine and cosine using the fundamental identities and then simplify the expression.

3. Show that (1 + cot2 θ)= is an identity.

Exercises

4. Simplify the following expressions.

a) (sec x + tan x)(sec x – tan x)

b) 2 –

Do Worksheet 6

Sum and Difference Identities

Double-Angle Identities

Sine Cosine Tangent

Half-Angle Identities

Sine Cosine Tangent

Product-to-Sum and Sum-to-Product Identities

Product-to-Sum Identities

Product-to-Sum and Sum-to-Product Identities

Sum-to-Product Identities

Examples 1. Find the exact value of sin 75° using sum

and difference identities.

2. Simplify sin 20°cos 40° + cos 20°sin 40°.

3. Simplify tan(x + 4π).

4. Given that cot θ = and θ is in the second quadrant, find:

a) sin 2θ b) tan 2θ c) cos 2θ

5. Find the exact value of sin 22.5° using half-angle identities.

6. Simplify cot (90° - θ) if tan θ = using cofunction identities.

7. Evaluate tan 165°.

8. Find the exact value of 105°.

9. Simplify cot ( - x ).

10. Simplify the following trigonometric expressions:

a) b)