Squaring a Sum
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 13 of 52.
Squaring a sum or difference of sine and cosine produces both squares, sin(x)² and cos(x)², and together they are 1. Take (sin(x) - cos(x))². Expanding gives sin(x)² - 2sin(x)cos(x) + cos(x)².
The two squares are not next to each other, but the order of terms in a sum does not matter, so the identity still applies to them: sin(x)² + cos(x)² becomes 1 and the board reads 1 - 2sin(x)cos(x).
The middle term is the trap. (a - b)² is not a² - b²: the cross term -2ab is part of the square, and here it is -2sin(x)cos(x).
Challenge
Show that (sin(x) + cos(x))² - 2sin(x)cos(x) equals 1: rewrite it until the board reads 1. Then press Check.
Try it yourself
Simplify the expression
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Trig Identities
1Identities on the Circle
sin²x + cos²x = 1Squares Erase the SignsIdentity or Equation?Recap: the Circle Identity4Even, Odd and Cofunctions
Negative AnglesEven and Odd in a ProofEven and Odd GraphsCofunctionsRecap: Signs of Angles2Reciprocal and Quotient
The Quotient IdentitySecant, Cosecant, CotangentDividing by a FractionWhere They BreakRecap: Reciprocals5Proving Identities
Work on One SideRewrite in sin and cosFactor FirstMake One FractionProofs That Go WrongRecap: Prove It8Half Angles, Power Reduction
Power Reductionsin²x Is a WaveHalf-Angle Valuestan of a Half AngleRecap: Half Angles3Pythagorean Identities
1 + tan²x = sec²x1 + cot²x = csc²xA Difference of SquaresSquaring a SumRecap: Three Forms