Verify the Identity

Problem

(1+cos(x))/(1−cos(x))−(1−cos(x))/(1+cos(x))=4*cot(x)*csc(x)

Solution

1. Find a common denominator for the fractions on the left side by multiplying the denominators (1−cos(x)) and (1+cos(x))

((1+cos(x))2−(1−cos(x))2)/((1−cos(x))*(1+cos(x)))

2. Expand the squares in the numerator using the identity (a±b)2=a2±2*a*b+b2

((1+2*cos(x)+cos2(x))−(1−2*cos(x)+cos2(x)))/((1−cos(x))*(1+cos(x)))

3. Simplify the numerator by distributing the negative sign and combining like terms.

(1+2*cos(x)+cos2(x)−1+2*cos(x)−cos2(x))/((1−cos(x))*(1+cos(x)))

(4*cos(x))/((1−cos(x))*(1+cos(x)))

4. Multiply the denominator using the difference of squares formula (a−b)*(a+b)=a2−b2

(4*cos(x))/(1−cos2(x))

5. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies 1−cos2(x)=sin2(x)

(4*cos(x))/sin2(x)

6. Rewrite the expression as a product of trigonometric functions to match the right side of the identity.

4⋅cos(x)/sin(x)⋅1/sin(x)

7. Substitute the definitions cot(x)=cos(x)/sin(x) and csc(x)=1/sin(x)

4*cot(x)*csc(x)

Final Answer

(1+cos(x))/(1−cos(x))−(1−cos(x))/(1+cos(x))=4*cot(x)*csc(x)

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