Simplify (sec(b)+csc(b))/(1+tan(b))

Problem

(sec(b)+csc(b))/(1+tan(b))

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the identities sec(b)=1/cos(b) csc(b)=1/sin(b) and tan(b)=sin(b)/cos(b)

(1/cos(b)+1/sin(b))/(1+sin(b)/cos(b))

2. Find a common denominator for the numerator and the denominator of the complex fraction.

(sin(b)+cos(b))/(cos(b)*sin(b))/(cos(b)+sin(b))/cos(b)

3. Multiply the numerator by the reciprocal of the denominator to simplify the complex fraction.

(sin(b)+cos(b))/(cos(b)*sin(b))⋅cos(b)/(cos(b)+sin(b))

4. Cancel the common factors sin(b)+cos(b) and cos(b) from the numerator and denominator.

1/sin(b)

5. Apply the reciprocal identity 1/sin(b)=csc(b)

csc(b)

Final Answer

(sec(b)+csc(b))/(1+tan(b))=csc(b)

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