Which of the following are trigonometric identities? Select all that apply.

Question

(2 points)
Responses

1. tangent theta equals Start Fraction sine theta over cosine theta End Fraction
Image with alt text: tangent theta equals Start Fraction sine theta over cosine theta End Fraction

2. cosine theta equals Start Fraction 1 over secant theta EnD Fraction
Image with alt text: cosine theta equals Start Fraction 1 over secant theta EnD Fraction

3. secant theta equals Start Fraction 1 over cosecant theta End Fraction
Image with alt text: secant theta equals Start Fraction 1 over cosecant theta End Fraction

4. sine squared theta minus cosine squared theta equals 1
Image with alt text: sine squared theta minus cosine squared theta equals 1

5. 1 plus tangent squared theta equals secant squared theta
Image with alt text: 1 plus tangent squared theta equals secant squared theta

Answer 1

Let's evaluate each of the provided options to determine if they are trigonometric identities:

1. tangent theta equals (sine theta / cosine theta)
   This is a valid trigonometric identity.
   Identity: True

2. cosine theta equals (1 / secant theta)
   This is also a valid trigonometric identity.
   Identity: True

3. secant theta equals (1 / cosecant theta)
   This is NOT a valid trigonometric identity.
   Identity: False (The correct relationship is secant theta = 1/cosine theta and cosecant theta = 1/sine theta)

4. sine squared theta minus cosine squared theta equals 1
   This is NOT a valid trigonometric identity. The correct identity is:
   \[ \sin^2 \theta + \cos^2 \theta = 1 \]
   Identity: False

5. 1 plus tangent squared theta equals secant squared theta
   This is a valid trigonometric identity (Pythagorean identity).
   Identity: True

Based on the evaluation, the identities are:

• 1: True
• 2: True
• 3: False
• 4: False
• 5: True

Thus, the selected responses that are trigonometric identities are:

1, 2, and 5.