Verify the Identity (cos(a+b))/(cos(a)sin(b))=cot(b)-tan(a)

Problem

cos(a+b)/(cos(a)*sin(b))=cot(b)−tan(a)

Solution

1. Apply the cosine sum identity to the numerator of the left side of the equation.

cos(a+b)=cos(a)*cos(b)−sin(a)*sin(b)

2. Substitute the expanded expression back into the fraction.

(cos(a)*cos(b)−sin(a)*sin(b))/(cos(a)*sin(b))

3. Split the fraction into two separate terms by dividing each part of the numerator by the common denominator.

(cos(a)*cos(b))/(cos(a)*sin(b))−(sin(a)*sin(b))/(cos(a)*sin(b))

4. Simplify each term by canceling common factors in the numerator and denominator.

cos(b)/sin(b)−sin(a)/cos(a)

5. Apply trigonometric definitions for cotangent and tangent to reach the final form.

cot(b)−tan(a)

Final Answer

cos(a+b)/(cos(a)*sin(b))=cot(b)−tan(a)

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