To prove the identity tan(A)·(2cos^3(A) - cos(A)) = sin(A) - 2sin^3(A), we will work on one side of the equation and manipulate it until it matches the other side. Here's the step-by-step explanation:
Starting with the left side of the equation:
tan(A)·(2cos^3(A) - cos(A))
1. We'll first simplify the expression inside the parentheses:
= 2cos^3(A) - cos(A)
2. Next, we'll rewrite tan(A) in terms of sin(A) and cos(A):
= (sin(A)/cos(A))(2cos^3(A) - cos(A))
3. Distribute the numerator (sin(A)) across the terms in the denominator (cos(A)):
= (2cos^3(A)sin(A)/cos(A)) - (cos(A)sin(A)/cos(A))
4. Simplify the expression further:
= 2cos^2(A)sin(A) - sin(A)
5. Now, we'll use the identity sin^2(A) + cos^2(A) = 1 and rewrite cos^2(A) as (1 - sin^2(A)):
= 2(1 - sin^2(A))sin(A) - sin(A)
6. Distribute the 2 across the terms:
= 2sin(A) - 2sin^3(A) - sin(A)
7. Combine like terms:
= sin(A) - 2sin^3(A)
We have now obtained the right side of the equation, which matches the simplified expression from the left side. Therefore, we have proved the identity:
tan(A)·(2cos^3(A) - cos(A)) = sin(A) - 2sin^3(A)