Prove the given
- (cot A - tan A)cos A = cosec A - 2sinA
Use identities
- sin²A + cos²A = 1
- cotA = cosA/sinA
- tanA = cosA/sinA
- cosecA = 1/sinA
Solution
Simplify the LHS by using the identities above to get the RHS:
- (cot A - tan A)cosA =
- (cosA/sinA - cosA/sinA)cosA = Identities 2 and 3
- (cosA/sinA)cosA - (cosA/sinA)cosA = Distribute
- cos²A/sinA - sinA = Simplify/cancel
- (1 - sin²A)/sinA - sinA = Identity 1
- 1/sinA - sin²A/sinA - sinA = Distribute
- 1/sinA - sinA - sinA = Simplify/cancel
- 1/sinA - 2sinA =
- cosecA - 2sinA Identity 4
Proved