Verify the Identity

Problem

(sec(x)−csc(x))/(sec(x)+csc(x))=(tan(x)−1)/(tan(x)+1)

Solution

1. Rewrite the left-hand side using the reciprocal identities sec(x)=1/cos(x) and csc(x)=1/sin(x)

(1/cos(x)−1/sin(x))/(1/cos(x)+1/sin(x))

2. Find a common denominator for the numerator and the denominator of the complex fraction, which is sin(x)*cos(x)

(sin(x)−cos(x))/(sin(x)*cos(x))/(sin(x)+cos(x))/(sin(x)*cos(x))

3. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

(sin(x)−cos(x))/(sin(x)+cos(x))

4. Divide every term in the numerator and the denominator by cos(x) to introduce the tangent function.

(sin(x)/cos(x)−cos(x)/cos(x))/(sin(x)/cos(x)+cos(x)/cos(x))

5. Apply the quotient identity tan(x)=sin(x)/cos(x) to reach the right-hand side.

(tan(x)−1)/(tan(x)+1)

Final Answer

(sec(x)−csc(x))/(sec(x)+csc(x))=(tan(x)−1)/(tan(x)+1)

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