Three Forms of cos 2x
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 33 of 52.
With B = A, cos(A + B) = cos(A)cos(B) - sin(A)sin(B) gives cos(2x) = cos(x)² - sin(x)². Replace one square using sin²x + cos²x = 1 and two more forms appear:
cos(2x) = 2cos(x)² - 1
cos(2x) = 1 - 2sin(x)²
All three are equal; choose the one that cancels something. In 1 + cos(2x) the form with -1 is the one to use: 1 + 2cos(x)² - 1, the ones cancel, and 2cos(x)² is left. In 1 - cos(2x) it is the form that starts with 1 that cancels.
Challenge
Show that cos(2x) + 2sin(x)² 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 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