Simplify (1+sin(y))/(1+csc(y))

Problem

(1+sin(y))/(1+csc(y))

Solution

1. Identify the trigonometric identity for the cosecant function.

csc(y)=1/sin(y)

2. Substitute the identity into the denominator of the original expression.

(1+sin(y))/(1+1/sin(y))

3. Find a common denominator for the terms in the denominator of the main fraction.

1+1/sin(y)=sin(y)/sin(y)+1/sin(y)

sin(y)/sin(y)+1/sin(y)=(sin(y)+1)/sin(y)

4. Rewrite the complex fraction using the simplified denominator.

(1+sin(y))/(sin(y)+1)/sin(y)

5. Multiply the numerator by the reciprocal of the denominator.

(1+sin(y))⋅sin(y)/(sin(y)+1)

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

((1+sin(y))⋅sin(y))/(1+sin(y))=sin(y)

Final Answer

(1+sin(y))/(1+csc(y))=sin(y)

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