Verify the Identity 1/(sec(x)-tan(x))=sec(x)+tan(x)

Problem

1/(sec(x)−tan(x))=sec(x)+tan(x)

Solution

1. Identify the left side of the equation to begin the verification process.

1/(sec(x)−tan(x))

2. Multiply the numerator and the denominator by the conjugate of the denominator, which is sec(x)+tan(x)

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

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

(sec(x)+tan(x))/(sec2(x)−tan2(x))

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

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

5. Simplify the expression to reach the right side of the identity.

sec(x)+tan(x)

Final Answer

1/(sec(x)−tan(x))=sec(x)+tan(x)

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