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Transform the first expression into the second.

A) (sec(x)-1)(sec(x)+1 to tan^2(x)
B) (1-sin(x))(1+sin(x)) to cos^2(x)
C) cos(x)(sec(x)+cos(x)csc^2(x)) to csc^2(x)
D) (1-tan(x))^2 to sec^2(x)-2tan(x)

1 Answer

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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.

User Christian Deger
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