The Quotient Identity
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 5 of 52.
On the circle the tangent is y over x, and the point is (cos(x), sin(x)). So
tan(x) = sin(x)/cos(x)
wherever cos x is not 0. Its flip is cot(x) = cos(x)/sin(x), wherever sin x is not 0. Rewriting tan and cot like this is often the first step of a proof, because it leaves only sines and cosines, and those can cancel.
Take cot(x)sin(x). Write cot as a quotient: cos(x)/sin(x) · sin(x). Multiply the tops and the bottoms: (cos(x)sin(x))/sin(x). The sin x on top and the sin x below cancel, and cos x is left. Cancelling divides by sin x, which is fine: where sin x is 0, cot x had no value to begin with.
Challenge
Show that tan(x)cos(x) equals sin(x): rewrite it until the board reads sin(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