1 + cot²x = csc²x
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 11 of 52.
Divide sin(x)² + cos(x)² = 1 by sin(x)² instead. The first term becomes 1, the second cot(x)², and the right side 1/sin(x)², which is csc(x)²:
1 + cot(x)² = csc(x)²
It holds wherever sin x is not 0. So there are three Pythagorean identities, one for each pair: sine and cosine, tangent and secant, cotangent and cosecant. The 1 always sits with tan² and cot², never with sec² or csc².
Take csc(x)² - cot(x)². Replace csc(x)² with 1 + cot(x)² and it reads 1 + cot(x)² - cot(x)². The cotangent squares cancel, and 1 is left.
Challenge
Show that (csc(x)² - 1)/cot(x) equals cot(x): rewrite it until the board reads cot(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 Angles3Pythagorean Identities
1 + tan²x = sec²x1 + cot²x = csc²xA Difference of SquaresSquaring a SumRecap: Three Forms