3.1 Trigonometric Identities

Goals

• Verify trigonometric identities.

A. Trigonometric Identities

Reciprocal Identities

• sin 𝑥 = 1/csc 𝑥

• csc 𝑥 = 1/sin 𝑥

• cos 𝑥 = 1/sec 𝑥

• sec 𝑥 = 1/cos 𝑥

• tan 𝑥 = 1/cot 𝑥

• cot 𝑥 = 1/tan 𝑥

Quotient Identities

• tan 𝑥 = sin 𝑥/cos 𝑥

• cot 𝑥 = cos 𝑥/sin 𝑥

Cofunction Identities

• sin(𝜋/2 − 𝑥) = cos 𝑥

• cos(𝜋/2 − 𝑥) = sin 𝑥

• tan(𝜋/2 − 𝑥) = cot 𝑥

• cot(𝜋/2 − 𝑥) = tan 𝑥

• sec(𝜋/2 − 𝑥) = csc 𝑥

• csc(𝜋/2 − 𝑥) = sec 𝑥

Pythagorean Identities

• sin² 𝑥 + cos² 𝑥 = 1

• tan² 𝑥 + 1 = sec² 𝑥

• 1 + cot² 𝑥 = csc² 𝑥

Even-Odd Identities

• sin(−𝑥) = −sin 𝑥

• csc(−𝑥) = −csc 𝑥

• cos(−𝑥) = cos 𝑥

• sec(−𝑥) = sec 𝑥

• tan(−𝑥) = −tan 𝑥

• cot(−𝑥) = −cot 𝑥

Verifying Identities

• Verifying means:

  • Making sure that both sides are equal.

  • Method is to manipulate one side to see if you can get the other.

• Methods to Use:

  1. Rewrite the expression in terms of sine and cosine only.

  2. Multiply a numerator and denominator of a ratio by a "well-chosen 1".

  3. Write sums of trig ratios as a single ratio.

  4. Factor.

  5. Multiply by the conjugate.

  6. Combine fractions with a common denominator.

B. Guidelines for Establishing Identities

1. Start with the side that contains the more complicated expression.

2. Rewrite sums or differences of quotients as a single quotient, i.e., a single fraction.

3. Rewrite one side in terms of sine and cosine functions only if helpful.

4. Keep in mind the goal while manipulating one side of the expression.

• It is customary to note the domains during verification.

Examples of Verification

1. Establish: csc 𝜃 ∙ tan 𝜃 = sec 𝜃

2. Establish: sin²(−𝜃) + cos²(−𝜃) = 1

3. Establish: (sin²(−𝜃) − cos²(−𝜃))/(sin(−𝜃) − cos(−𝜃)) = cos 𝜃 − sin 𝜃

Practice Examples

Example 1

• Show: 1 + tan𝜃 = 1 + cot𝜃

Example 2

• Show: sin 𝜃/(1+cos𝜃) + (1+cos𝜃)/sin𝜃 = 2 csc 𝜃

  • Rewrite in terms of sine and cosine only.

  • Obtain a common denominator.

Example 3

• Show: tan 𝜃 + cot𝜃 = sec𝜃 + csc𝜃

Example 4

• Show: 1 - sin𝜃 = cos𝜃(1 + sin𝜃)

  • Rewrite with sines and cosines.

Example 5

• Show: cos𝜃 cot𝜃/(1 - sin𝜃) - 1 = csc 𝜃

Example 6

• Show: 1/(sec𝜃 tan𝜃) = csc 𝜃 - sin 𝜃