Verify the Identity tan(x)+(cos(x))/(1+sin(x))=1/(cos(x))

Problem

tan(x)+cos(x)/(1+sin(x))=1/cos(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 this identity into the left side of the equation to prepare for finding a common denominator.

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

3. Find a common denominator by multiplying the first term by (1+sin(x))/(1+sin(x)) and the second term by cos(x)/cos(x)

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

4. Distribute the sine term in the numerator.

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

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

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

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

1/cos(x)

Final Answer

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

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