Question

A) Simplify the expression (1 + cos(x))(1 - cos(x)).

A. 1 - cos^2(x)
B. 1
C. sin^2(x)
D. cos^2(x)

b) Simplify the expression 1/tan^2(x) - 1/cos^2(x).

A. sin^2(x) - 1
B. cos^2(x) - 1
C. tan^2(x)
D. 1 - sin^2(x)

c) Simplify the expression sec^2(π/2 - x)[sin^2(x) - sin^4(x)].

A. cos^2(x)
B. sin^2(x)
C. tan^2(x)
D. cot^2(x)

Answer 1

To simplify the given trigonometric expressions: (a) (1 + cos(x))(1 - cos(x)) simplifies to sin^2(x), (b) 1/tan^2(x) - 1/cos^2(x) simplifies to sin^2(x) - 1, and (c) sec^2(π/2 - x)[sin^2(x) - sin^4(x)] simplifies to sin^2(x).

Step-by-step explanation:

To answer the student's questions on simplifying trigonometric expressions:

1. Simplify the expression (1 + cos(x))(1 - cos(x)). The correct answer is C. sin^2(x), because this is a difference of squares which simplifies to 1 - cos^2(x), and according to the Pythagorean identity, sin^2(x) = 1 - cos^2(x).
2. Simplify the expression 1/tan^2(x) - 1/cos^2(x). The correct answer is A. sin^2(x) - 1 because 1/tan^2(x) is equal to cos^2(x)/sin^2(x), and when subtracting 1/cos^2(x) from this, we get (cos^2(x) - 1)/sin^2(x), which simplifies to sin^2(x) - 1.
3. Simplify the expression sec^2(π/2 - x)[sin^2(x) - sin^4(x)]. The correct answer is B. sin^2(x) because sec^2(π/2 - x) is equal to 1/cos^2(π/2 - x), which is the same as tan^2(x), and when multiplied by sin^2(x) - sin^4(x), the tan^2(x) and sin^4(x) cancel out, leaving sin^2(x).