To confirm the identity (1 - cos(x)) = sin^2(x) and (csc(x) - tan(x))(csc(x) + tan(x)) = 1, we can simplify each expression separately:
1. Confirming the identity (1 - cos(x)) = sin^2(x):
Starting with the left-hand side (LHS):
LHS = 1 - cos(x)
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite the right-hand side (RHS):
sin^2(x) = 1 - cos^2(x)
Now, let's compare the LHS and RHS:
LHS = 1 - cos(x)
RHS = sin^2(x) = 1 - cos^2(x)
Since both the LHS and RHS are equal, we have confirmed the identity (1 - cos(x)) = sin^2(x).
2. Confirming the identity (csc(x) - tan(x))(csc(x) + tan(x)) = 1:
Starting with the left-hand side (LHS):
LHS = (csc(x) - tan(x))(csc(x) + tan(x))
To simplify, we'll use the reciprocal and quotient identities:
csc(x) = 1 / sin(x)
tan(x) = sin(x) / cos(x)
Substituting these identities into the expression:
LHS = (1 / sin(x) - sin(x) / cos(x))(1 / sin(x) + sin(x) / cos(x))
To simplify further, we can find a common denominator:
LHS = [(cos(x) - sin^2(x)) / (sin(x) * cos(x))][(cos(x) + sin^2(x)) / (sin(x) * cos(x))]
Now, let's multiply the numerators:
LHS = (cos^2(x) - sin^4(x)) / (sin^2(x) * cos^2(x))
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite the numerator:
LHS = (cos^2(x) - (1 - cos^2(x))^2) / (sin^2(x) * cos^2(x))
LHS = (cos^2(x) - (1 - 2cos^2(x) + cos^4(x))) / (sin^2(x) * cos^2(x))
LHS = (3cos^2(x) - cos^4(x) - 1) / (sin^2(x) * cos^2(x))
Simplifying further, we have:
LHS = (3cos^2(x) - cos^4(x) - 1) / (sin^2(x) * cos^2(x))
Since the expression does not simplify to 1, the identity (csc(x) - tan(x))(csc(x) + tan(x)) = 1 does not hold true.
Therefore, only the first identity (1 - cos(x)) = sin^2(x) is confirmed, while the second identity is not valid.