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, which is cos(x)*(1+sin(x))

2. Rewrite the fractions by multiplying the numerator and denominator of each term by the missing factor from the common denominator.

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

3. Expand the numerator by squaring the binomial (1+sin(x))2

(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 constant 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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