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

Problem

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

Solution

1. Start with the right side of the identity to simplify the expression toward the left side.

(sec(x)−tan(x))^2

2. Rewrite the trigonometric functions in terms of sine and cosine using the reciprocal and quotient identities.

(1/cos(x)−sin(x)/cos(x))^2

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

((1−sin(x))/cos(x))^2

4. Apply the power to both the numerator and the denominator.

((1−sin(x))^2)/cos^2(x)

5. Use the Pythagorean identity cos^2(x)=1−sin^2(x) to rewrite the denominator.

((1−sin(x))^2)/(1−sin^2(x))

6. Factor the denominator as a difference of squares, where 1−sin^2(x)=(1−sin(x))*(1+sin(x))

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

7. Cancel the common factor (1−sin(x)) from the numerator and the denominator to reach the left side of the identity.

(1−sin(x))/(1+sin(x))

Final Answer

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

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