Rewrite in sin and cos
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 21 of 52.
When no identity fits the expression as it stands, write every function in sines and cosines. It is rarely the shortest route, but it always gives something to work with.
Prove cot(x)²sin(x)² + sin(x)² = 1. Write cot x as cos(x)/sin(x), so the first term is (cos(x)/sin(x))² · sin(x)². Square the fraction, cos(x)²/sin(x)², multiply the tops and the bottoms, and cancel sin(x)²: the first term is now cos(x)². The line reads cos(x)² + sin(x)², which is 1.
Squaring a quotient squares its top and its bottom: (a/b)² = a²/b². Forgetting to square the bottom is the slip to watch for.
Challenge
Prove that cos(x)² + cos(x)²tan(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 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