Final answer:
To verify the trigonometric identities, we need to simplify each equation and check if they hold true for all values of theta. The first equation simplifies to a true statement, making it a trigonometric identity. However, the second equation simplifies to a false statement, indicating that it is not an identity.Option C is the correct answer.
Step-by-step explanation:
Let's analyze each equation to determine if they are trigonometric identities:
1+ cos²θ /cot²θ (1-sin²θ ) =12sec²θ:
First, replace cot²θ with the equivalent expression 1/tan²θ = cos²θ/ sin²θ. Simplifying the equation, we get (1 + cos²θ)/(cos²θ/sin²θ) * (1 - sin²θ) = 12/cos²θ. The left side can be simplified further by canceling out the common factors, leaving sin²θ * (1 - sin²θ) = 12/cos²θ. On the right side, we convert sec²θ to 1/cos²θ.
The equation thus becomes sin²θ - sin⁴θ = 12/cos²θ. Since the equation can be simplified to a true statement, it is a trigonometric identity.
18cos θ ( 1/cos θ - cot θ /csc θ )=18sin²θ:
Start by simplifying the expression in the brackets: 1/cos θ - cot θ /csc θ = 1/cos θ - cos θ /sin θ /sin θ = (sin θ - cos²θ)/sin²θ. Now, substitute this expression back into the original equation and simplify: 18cos θ * (sin θ - cos²θ)/sin²θ = 18sin²θ.
Cross-multiplying and simplifying, we get 18cos θ * (sin θ - cos²θ) = 18sin²θ * sin²θ. Both sides can be further simplified, resulting in a false statement. Therefore, the second equation is not a trigonometric identity.
Based on this analysis, only the first equation 1+ cos²θ /cot²θ (1-sin²θ ) =12sec²θ is a trigonometric identity. Therefore, the correct answer is option c. Only the first equation is an identity.