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.

RHS=(sec(x)+tan(x))^2

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

sec(x)=1/cos(x)

tan(x)=sin(x)/cos(x)

3. Substitute these identities into the expression and combine the terms over a common denominator.

RHS=(1/cos(x)+sin(x)/cos(x))^2

RHS=((1+sin(x))/cos(x))^2

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

RHS=((1+sin(x))^2)/cos^2(x)

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

RHS=((1+sin(x))^2)/(1−sin^2(x))

6. Factor the denominator as a difference of squares.

1−sin^2(x)=(1−sin(x))*(1+sin(x))

7. Simplify the fraction by canceling the common factor (1+sin(x)) from the numerator and the denominator.

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

RHS=(1+sin(x))/(1−sin(x))

Final Answer

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

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