Prove the following using trigonometric identities:1. (1 - cos2θ) csc2θ = 12. sec θ √(1 - sin^2θ) = 13. tan θ sin θ + cos θ = sec θ4. (1 - cos θ)(1 + cos θ)(1 + cot^2θ) = 15. cot θ + tan θ = sec θ csc θ

To prove the given trigonometric identities, we will use various trigonometric identities and properties. Let's start with each identity one by one:

1. (1 - cos^2θ) csc^2θ = 1 / sin^2θ Using the Pythagorean identity sin^2θ + cos^2θ = 1, we can rewrite the left side of the equation as (sin^2θ / sin^2θ) = 1 / sin^2θ. Therefore, the left side is equal to the right side, proving the identity.

2. secθ √(1 - sin^2θ) = 1 / cosθ √(1 - sin^2θ) Using the Pythagorean identity sin^2θ + cos^2θ = 1, we can rewrite the expression inside the square root as cos^2θ. Therefore, the expression becomes 1 / cosθ √cos^2θ, which simplifies to 1 / cosθ * cosθ = 1, proving the identity.

3. tanθ sinθ + cosθ = (sinθ / cosθ) * sinθ + cosθ Simplifying the expression on the left side, we get sin^2θ / cosθ + cosθ. Using the identity sin^2θ + cos^2θ = 1, we can rewrite the expression as (1 - cos^2θ) / cosθ + cosθ. Simplifying further, we get (1 + cos^2θ) / cosθ = secθ, proving the identity.

4. (1 - cosθ)(1 + cosθ)(1 + cot^2θ) = (1 - cos^2θ)(1 + cot^2θ) Expanding both sides of the equation, we get (1 - cos^2θ - cosθ + cos^3θ)(1 + cot^2θ). Simplifying further, we get (sin^2θ - cosθ + cos^3θ)(1 + cot^2θ). Using the identity sin^2θ = 1 - cos^2θ, we can rewrite the expression as (1 - cos^2θ - cosθ + cos^3θ)(1 + cot^2θ). Simplifying further, we get (1 - cosθ)(1 + cosθ)(1 + cot^2θ), which is the same as the right side of the equation, proving the identity.

5. cotθ + tanθ = cosθ / sinθ + sinθ / cosθ Combining the fractions on the right side, we get (cos^2θ + sin^2θ) / (sinθ * cosθ). Using the identity sin^2θ + cos^2θ = 1, we can rewrite the expression as 1 / (sinθ * cosθ). Simplifying further, we get cscθ * secθ, which is equal to secθ * cscθ, proving the identity.

Therefore, we have successfully proven all the given trigonometric identities.