Dividing by a Fraction
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 7 of 52.
When the bottom of a fraction is itself a fraction, flip it and multiply: dividing by 2/3 is multiplying by 3/2. The same rule works with trig functions.
Take sin(x)/tan(x). Write the tangent as a quotient: sin(x)/(sin(x)/cos(x)). Multiply by the reciprocal: sin(x) · (cos(x)/sin(x)). Multiply the tops and the bottoms, (sin(x)cos(x))/sin(x), and cancel sin x. What is left is cos x.
Every step holds wherever the start has a value. Where tan x is 0 or undefined the start has none, so the identity says nothing there.
Challenge
Show that sec(x)/tan(x) equals csc(x): rewrite it until the board reads csc(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