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

Problem

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

Solution

1. Identify the side to manipulate. We will start with the right-hand side (RHS) and simplify it to match the left-hand side (LHS).

2. Rewrite the trigonometric functions in terms of sine and cosine using the reciprocal and quotient identities.

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

3. Combine the terms inside the parentheses since they share a common denominator.

((1−cos(x))/sin(x))2

4. Apply the power to both the numerator and the denominator.

((1−cos(x))2)/sin2(x)

5. Substitute the Pythagorean identity sin2(x)=1−cos2(x) into the denominator.

((1−cos(x))2)/(1−cos2(x))

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

((1−cos(x))*(1−cos(x)))/((1−cos(x))*(1+cos(x)))

7. Cancel the common factor (1−cos(x)) from the numerator and the denominator.

(1−cos(x))/(1+cos(x))

Final Answer

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

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