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

Problem

1/(sec(x)−tan(x))−1/(sec(x)+tan(x))

Solution

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

Denominator=(sec(x)−tan(x))*(sec(x)+tan(x))

2. Rewrite the expression over the common denominator.

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

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

sec(x)+tan(x)−sec(x)+tan(x)=2*tan(x)

4. Simplify the denominator using the difference of squares formula.

(sec(x)−tan(x))*(sec(x)+tan(x))=sec2(x)−tan2(x)

5. Apply the Pythagorean identity sec2(x)−tan2(x)=1 to the denominator.

(2*tan(x))/1=2*tan(x)

Final Answer

1/(sec(x)−tan(x))−1/(sec(x)+tan(x))=2*tan(x)

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