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.

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

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

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

2/((sec(x)−1)*(sec(x)+1))

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

2/(sec2(x)−1)

4. Apply a Pythagorean identity using the relation tan2(x)+1=sec2(x) which implies sec2(x)−1=tan2(x)

2/tan2(x)

5. Use a reciprocal identity to rewrite the expression, since 1/tan(x)=cot(x)

2*cot2(x)

Final Answer

1/(sec(x)−1)−1/(sec(x)+1)=2*cot2(x)

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