Prove that the equation below is an identity:
2 sin2Ө – sin4Ө = (1 – cos2Ө)(1 + cos2Ө).
To prove that an equation is an identity, one must show that the Left Hand Side (LHS) of the equation is identical to its Right Hand Side (RHS). That is, LHS ≡ RHS.
LHS: 2 sin2Ө – sin4Ө = sin2Ө (2 – sin2Ө)
= (1 – cos2Ө)(2-(1 – cos2Ө)). This is because sin2Ө=(1 – cos2Ө) from the Pythagorean identity sin2Ө+cos2Ө=1.
= (1 – cos2Ө)(2-1+ cos2Ө). Remember the rule about removing signs preceded by a minus sign.
= (1 – cos2Ө)(1 + cos2Ө)
LHS ≡ RHS. QED.
Note: This post is actually for my nephew. I hope the teacher will let him explain this. It should lessen my guilt as math teacher … and I should buy myself a stylus.
Click Proving trigonometric identities #1 for another example.