Verify the Identity (sec(x)-tan(x))^2=(1-sin(x))/(1+sin(x))

Problem

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

Solution

1. Rewrite the terms inside the parentheses using the fundamental identities sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x)

(sec(x)−tan(x))2=(1/cos(x)−sin(x)/cos(x))2

2. Combine the fractions inside the parentheses since they share a common denominator.

((1−sin(x))/cos(x))2

3. Distribute the square to both the numerator and the denominator.

((1−sin(x))2)/cos2(x)

4. Substitute the Pythagorean identity cos2(x)=1−sin2(x) into the denominator.

((1−sin(x))2)/(1−sin2(x))

5. Factor the denominator as a difference of squares using the formula a2−b2=(a−b)*(a+b)

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

6. Cancel the common factor (1−sin(x)) from the numerator and the denominator to reach the final form.

(1−sin(x))/(1+sin(x))

Final Answer

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

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