Verify the Identity

Problem

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

Solution

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

(cos(x)⋅cos(x)+(1+sin(x))*(1+sin(x)))/(cos(x)*(1+sin(x)))

2. Expand the numerator by multiplying the terms.

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

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

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

4. Combine like terms in the numerator.

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

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

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

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

2/cos(x)

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

2*sec(x)

Final Answer

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

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