Feb 232013
 
Problem:

Prove that the equation below is an identity:

2 sin2Ө – sin4Ө = (1 – cos2Ө)(1 + cos2Ө).

Solution

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.

Proof

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)

  = RHS

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.

proving identities

Click Proving trigonometric identities #1 for another example.

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