Final answer:
To establish the identity cosθ(tanθ cotθ)=csc θ, we can manipulate the left side of the equation using trigonometric identities, and simplify the right side of the equation. Using the identity tan θ = sin θ / cos θ and cot θ = 1/tan θ = cos θ / sin θ, we can rewrite the left side of the equation as cos θ (sin θ / cos θ) (cos θ / sin θ), which simplifies to 1/sin θ. Therefore, the identity is established.
Step-by-step explanation:
To establish the identity cosθ(tanθ cotθ)=csc θ, we can manipulate the left side of the equation using trigonometric identities. Let's start by simplifying the right side of the equation.
We know that csc θ = 1/sin θ. Therefore, the right side can be written as 1/sin θ. Now let's simplify the left side of the equation.
Using the identity tan θ = sin θ / cos θ and cot θ = 1/tan θ = cos θ / sin θ, we can rewrite the left side of the equation as cos θ (sin θ / cos θ) (cos θ / sin θ). This expression simplifies to 1/sin θ, which is equal to the right side of the equation. Therefore, the identity is established.