Proving the Identity: sin A - 2sin³(A/2) / (cos³A - cosA = tan A
To prove the given identity, we'll break it down step by step.
Step 1: Simplifying the Expression
- Start with the left-hand side:
sin A - 2sin³(A/2) / (cos³A - cosA)
- Use the identity sin A = 2sin(A/2)cos(A/2) to express sin A in terms of sin(A/2).
Step 2: Rewrite the Cubic Terms
- Notice that cos³A can be factored:
cos³A - cosA = cosA(cos²A - 1)
This can be further simplified using the identity cos²A = 1 - sin²A.
Step 3: Substitute and Factor
- After substituting, you can express all terms in terms of sin(A/2) and cos(A/2).
- Factor out common terms to simplify the expression.
Step 4: Relate to tan A
- Recall that tan A = sin A / cos A.
- By substituting in the simplified expression from the previous steps, you can show that the simplified expression equals tan A.
Conclusion
- After careful manipulation and substitution of trigonometric identities, you will find that the left-hand side simplifies to the right-hand side, confirming the identity:
sin A - 2sin³(A/2) / (cos³A - cosA) = tan A.
This methodical approach demonstrates the equality and proves the identity as required.