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

Problem

1/(1−cos(x))+1/(1+cos(x))=2*csc^2(x)

Solution

1. Find a common denominator for the two fractions on the left side of the equation.

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

2. Simplify the numerator by combining like terms.

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

2/((1−cos(x))*(1+cos(x)))

3. Expand the denominator using the difference of squares formula (a−b)*(a+b)=a^2−b^2

2/(1−cos^2(x))

4. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 which implies 1−cos^2(x)=sin^2(x)

2/sin^2(x)

5. Use the reciprocal identity csc(x)=1/sin(x) to rewrite the expression.

2*csc^2(x)

Final Answer

1/(1−cos(x))+1/(1+cos(x))=2*csc^2(x)

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