Question

Q6 Q21.) Verify the identity, write the left side numerator in terms of a sum or difference formula for sine or cosine, rewrite the expression found in the previous step by separating the denominator, and the expression from the previous step then simplifies to cot α - tan β using​ what?

Answer 1

Expanding cos(alpha+beta) = cos(alpha)*cos(beta) - sin(alpha)*sin(beta)

Separating the denominator

cos(alpha+beta)/[sin(alpha)*cos(beta)] = cos(alpha)*cos(beta)/[sin(alpha)*cos(beta)] - sin(alpha)*sin(beta)/[sin(alpha)*cos(beta)]

Using Quotient Identity, the expression becomes

cos(alpha+beta)/[sin(alpha)*cos(beta)] = cos(alpha)/sin(alpha) - sin(beta)/cos(beta)

= cot(alpha) - tan(beta)

Answer 2

cos(α + β) = cosα*cosβ - sinα*sinβ

denominator remains the same for both expressions as sinα*cosβ

the simplification uses A. Quotient Identity