To transform the given expressions into the desired form, we'll use various trigonometric identities:
A) (sec(x) - 1)(sec(x) + 1) to tan^2(x)
We can use the identity sec^2(x) - 1 = tan^2(x).
Using this identity:
(sec(x) - 1)(sec(x) + 1) = sec^2(x) - 1 = tan^2(x)
Therefore, the expression (sec(x) - 1)(sec(x) + 1) can be transformed into tan^2(x).
B) (1 - sin(x))(1 + sin(x)) to cos^2(x)
Using the identity 1 - sin^2(x) = cos^2(x), we can simplify the expression:
(1 - sin(x))(1 + sin(x)) = 1 - sin^2(x) = cos^2(x)
Therefore, the expression (1 - sin(x))(1 + sin(x)) can be transformed into cos^2(x).
C) cos(x)(sec(x) + cos(x)csc^2(x)) to csc^2(x)
Using the identity sec(x) = 1/cos(x) and csc(x) = 1/sin(x), we can simplify the expression:
cos(x)(sec(x) + cos(x)csc^2(x)) = cos(x)(1/cos(x) + cos(x)(1/sin^2(x))) = 1 + cos^2(x)/sin^2(x)
Now, using the identity 1 + cot^2(x) = csc^2(x), we can further simplify:
1 + cos^2(x)/sin^2(x) = 1 + cot^2(x) = csc^2(x)
Therefore, the expression cos(x)(sec(x) + cos(x)csc^2(x)) can be transformed into csc^2(x).
D) (1 - tan(x))^2 to sec^2(x) - 2tan(x)
Expanding the square and simplifying, we get:
(1 - tan(x))^2 = 1 - 2tan(x) + tan^2(x) = sec^2(x) - 2tan(x)
Therefore, the expression (1 - tan(x))^2 can be transformed into sec^2(x) - 2tan(x).
By using the appropriate trigonometric identities, we have transformed the given expressions into the desired forms.