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

Problem

1/(sec(x)+1)−1/(sec(x)−1)

Solution

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

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

2. Expand the numerator by distributing the negative sign.

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

3. Simplify the numerator by combining like terms.

(−2)/((sec(x)+1)*(sec(x)−1))

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

(−2)/(sec2(x)−1)

5. Apply the Pythagorean identity tan2(x)+1=sec2(x) which implies sec2(x)−1=tan2(x)

(−2)/tan2(x)

6. Use the reciprocal identity cot(x)=1/tan(x) to rewrite the expression.

−2*cot2(x)

Final Answer

1/(sec(x)+1)−1/(sec(x)−1)=−2*cot2(x)

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