A Difference of Squares
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 12 of 52.
(a - b)(a + b) = a² - b², and when a is 1 that difference is often a Pythagorean identity in disguise. Take (1 - sin(x))(1 + sin(x)). Expand it: 1 + sin(x) - sin(x) - sin(x)². The middle terms cancel and 1 - sin(x)² is left, which is cos(x)².
The same works for secant and cosecant, whose squares are 1 more than another square. (csc(x) - 1)(csc(x) + 1) expands to csc(x)² - 1, which is cot(x)². So a product of a function minus one and the same function plus one is worth expanding when its square minus 1 is an identity: it lands on a single square.
Challenge
Show that (sec(x) - 1)(sec(x) + 1) equals tan(x)²: rewrite it until the board reads tan(x)². 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