Even and Odd in a Proof
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 16 of 52.
When a negative angle sits inside a proof, clear it first: the rest of the work then runs in angle x alone. Take cos(-x)tan(x). Cosine is even, so it becomes cos(x)tan(x). Write tan as a quotient, cos(x) · (sin(x)/cos(x)), multiply the tops and the bottoms, and cancel cos x. What is left is sin x.
With an odd function the minus comes out in front and rides along to the end: sin(-x)csc(x) is -sin(x)csc(x), which is -1. Drop that minus and the last line is wrong.
Challenge
Show that sin(-x)cot(x) equals -cos(x): rewrite it until the board reads -cos(x). 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 Angles