Prove the following Identities
(1-cos^2X)csc^2X=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.
Prove the following trigonometry identity
2.𝑠𝑒𝑐𝜃√(1 − 𝑠𝑖𝑛2𝜃) = 1
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.
To prove the identity (1-cos^2X)csc^2X = 1, we will start by simplifying the left-hand side (LHS) of the equation:
LHS = (1 - cos^2X) * csc^2X
Recall the Pythagorean identity for sine and cosine:
sin^2X + cos^2X = 1
Rearranging this identity, we have:
sin^2X = 1 - cos^2X
Substituting this into our equation:
LHS = (1 - cos^2X) * csc^2X
= sin^2X * csc^2X
Next, recall the reciprocal identities for sine and cosecant:
cscX = 1/sinX
Substituting this into our equation:
LHS = sin^2X * (1/sinX)^2
= sin^2X * (1/sinX) * (1/sinX)
= sin^2X/sinX * 1/sinX
Now, simplifying the expression:
LHS = sin^2X/sinX * 1/sinX
= (sinX * sinX)/sinX * 1/sinX
Cancelling out the common factor of sinX in the numerator and denominator:
LHS = sinX * 1
= sinX
Finally, we can see that the LHS simplifies to sinX, which is equal to the right-hand side (RHS) of the equation. Thus, we have proved that:
(1 - cos^2X) * csc^2X = 1