Simplify (cos(x))/(1+sin(x))+tan(x)

Problem

cos(x)/(1+sin(x))+tan(x)

Solution

1. Rewrite the tangent function in terms of sine and cosine using the quotient identity.

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

2. Substitute the identity back into the original expression.

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

3. Find a common denominator to combine the two fractions, which is cos(x)*(1+sin(x))

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

4. Combine the numerators over the common denominator.

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

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

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

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

1/cos(x)

7. Apply the reciprocal identity to write the final result in terms of the secant function.

sec(x)

Final Answer

cos(x)/(1+sin(x))+tan(x)=sec(x)

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