Verify the Identity

Problem

(1−sin(x))/cos(x)+cos(x)/(1−sin(x))=2*sec(x)

Solution

1. Find a common denominator for the two fractions on the left side of the equation.

Common Denominator=cos(x)*(1−sin(x))

2. Rewrite the fractions using the common denominator by multiplying the first fraction by (1−sin(x))/(1−sin(x)) and the second by cos(x)/cos(x)

((1−sin(x))2+cos2(x))/(cos(x)*(1−sin(x)))

3. Expand the numerator by squaring the binomial (1−sin(x))

(1−2*sin(x)+sin2(x)+cos2(x))/(cos(x)*(1−sin(x)))

4. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to simplify the numerator.

(1−2*sin(x)+1)/(cos(x)*(1−sin(x)))

5. Combine like terms in the numerator.

(2−2*sin(x))/(cos(x)*(1−sin(x)))

6. Factor out the common factor of 2 from the numerator.

(2*(1−sin(x)))/(cos(x)*(1−sin(x)))

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

2/cos(x)

8. Apply the reciprocal identity 1/cos(x)=sec(x) to reach the final form.

2*sec(x)

Final Answer

(1−sin(x))/cos(x)+cos(x)/(1−sin(x))=2*sec(x)

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