Prove the following Identities

Question

(1-cos^2X)csc^2X=1

Answer 1

To prove the identity (1 - cos^2(X))csc^2(X) = 1, we will start with the left-hand side (LHS) and simplify it until it equals 1.

LHS: (1 - cos^2(X))csc^2(X)

Using the Pythagorean Identity sin^2(X) + cos^2(X) = 1, we can rewrite cos^2(X) as 1 - sin^2(X):

LHS: (1 - (1 - sin^2(X)))csc^2(X)

Simplifying the expression inside the parentheses:

LHS: (1 - 1 + sin^2(X))csc^2(X)

LHS: sin^2(X)csc^2(X)

Using the reciprocal identity csc(X) = 1/sin(X):

LHS: sin^2(X)(1/sin^2(X))

The sin^2(X) term cancels out:

LHS: 1

Therefore, the left-hand side (LHS) simplifies to 1, proving the given identity.

Answer 2

Prove the following trigonometry identity
2.𝑠𝑒𝑐𝜃√(1 − 𝑠𝑖𝑛2𝜃) = 1

Answer 3

To prove the identity 2secθ√(1 - sin^2θ) = 1, we will start with the left-hand side (LHS) and simplify it until it equals 1.

LHS: 2secθ√(1 - sin^2θ)

Using the Pythagorean Identity sin^2θ + cos^2θ = 1, we can rewrite 1 - sin^2θ as cos^2θ:

LHS: 2secθ√(cos^2θ)

The square root of cos^2θ is simply cosθ:

LHS: 2secθcosθ

Recall the definition of secθ as 1/cosθ:

LHS: 2(1/cosθ)cosθ

The cosθ terms cancel out:

LHS: 2

Therefore, the left-hand side (LHS) simplifies to 2, not 1. This means that the given identity is not true.