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

Problem

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

Solution

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

cos(a−b)=cos(a)*cos(b)+sin(a)*sin(b)

2. Substitute this identity back into the original expression.

(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 out common factors in the numerator and denominator.

cos(b)/sin(b)+sin(a)/cos(a)

5. Use trigonometric definitions to rewrite the ratios as cotangent and tangent functions.

cot(b)+tan(a)

6. Rearrange the terms to match the right side of the original identity.

tan(a)+cot(b)

Final Answer

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

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