Verify the Identity (csc(x)-cot(x))^2=(1-cos(x))/(1+cos(x))

Problem

(csc(x)−cot(x))2=(1−cos(x))/(1+cos(x))

Solution

1. Rewrite the terms inside the parentheses using the fundamental trigonometric identities csc(x)=1/sin(x) and cot(x)=cos(x)/sin(x)

(1/sin(x)−cos(x)/sin(x))2

2. Combine the fractions inside the parentheses since they share a common denominator.

((1−cos(x))/sin(x))2

3. Distribute the square to both the numerator and the denominator.

((1−cos(x))2)/sin2(x)

4. Apply the Pythagorean identity sin2(x)=1−cos2(x) to the denominator.

((1−cos(x))2)/(1−cos2(x))

5. Factor the denominator as a difference of squares, where 1−cos2(x)=(1−cos(x))*(1+cos(x))

((1−cos(x))2)/((1−cos(x))*(1+cos(x)))

6. Simplify the expression by canceling the common factor (1−cos(x)) from the numerator and the denominator.

(1−cos(x))/(1+cos(x))

Final Answer

(csc(x)−cot(x))2=(1−cos(x))/(1+cos(x))

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