Simplify 1/(1+cos(x))-1/(1-cos(x))

Problem

1/(1+cos(x))−1/(1−cos(x))

Solution

1. Find a common denominator by multiplying the two denominators together.

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

2. Expand the numerator by distributing the negative sign.

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

3. Simplify the numerator by combining like terms.

(−2*cos(x))/((1+cos(x))*(1−cos(x)))

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

(−2*cos(x))/(1−cos2(x))

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

(−2*cos(x))/sin2(x)

6. Rewrite the expression using the definitions of cotangent and cosecant, where cos(x)/sin(x)=cot(x) and 1/sin(x)=csc(x)

−2⋅cos(x)/sin(x)⋅1/sin(x)

7. Finalize the simplification by substituting the trigonometric functions.

−2*cot(x)*csc(x)

Final Answer

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

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