Square, Then Double
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 35 of 52.
Squaring a sine and a cosine together gives the two squares, which make 1, and a cross term 2 sin x cos x, which is sin 2x. Take (sin(x) - cos(x))². Expand it: sin(x)² - 2sin(x)cos(x) + cos(x)². Fold the cross term into -sin(2x) and the squares into 1, and the line reads 1 - sin(2x).
That says something about sizes: a square is never negative, so 1 - sin 2x is never negative, which is why sin 2x is never more than 1.
Challenge
Show that (cos(x) + sin(x))² - 1 equals sin(2x): rewrite it until the board reads sin(2x). 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 Angles7Double-Angle Formulas
sin 2x = 2 sin x cos xThree Forms of cos 2xExact Values, DoubledSquare, Then DoubleThe Graph of 2 sin x cos xRecap: Double 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 Angles