Simplify (sec(x)-tan(x))^2

Problem

(sec(x)−tan(x))2

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the identities sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x)

(1/cos(x)−sin(x)/cos(x))2

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

((1−sin(x))/cos(x))2

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

((1−sin(x))2)/cos2(x)

4. Apply the Pythagorean identity cos2(x)=1−sin2(x) to the denominator.

((1−sin(x))2)/(1−sin2(x))

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

((1−sin(x))2)/((1−sin(x))*(1+sin(x)))

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

(1−sin(x))/(1+sin(x))

Final Answer

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

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