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

Problem

1/(csc(x)−1)−1/(csc(x)+1)

Solution

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

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

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

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

2/((csc(x)−1)*(csc(x)+1))

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

2/(csc2(x)−1)

4. Apply the Pythagorean identity cot2(x)+1=csc2(x) which implies csc2(x)−1=cot2(x)

2/cot2(x)

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

2*tan2(x)

Final Answer

1/(csc(x)−1)−1/(csc(x)+1)=2*tan2(x)

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