Sums Over Sums
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 45 of 52.
A fraction with a sum on top and a sum below often turns into two products that share a factor. Take (sin(5x) + sin(3x))/(cos(5x) + cos(3x)). The top is 2sin(4x)cos(x) and the bottom is 2cos(4x)cos(x). The 2 and the cos x cancel, and sin(4x)/cos(4x) is tan(4x).
Turn both into products first, then look for the common factor. A sum cannot be cancelled term by term; a product can.
Challenge
Prove that (sin(3x) + sin(x))/(cos(3x) + cos(x)) equals tan(2x): rewrite it until the board reads tan(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 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