1 + tan²x = sec²x
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 10 of 52.
Divide every term of sin(x)² + cos(x)² = 1 by cos(x)². The first term becomes tan(x)², the second becomes 1, and the right side becomes 1/cos(x)², which is sec(x)²:
1 + tan(x)² = sec(x)²
It holds wherever cos x is not 0, the same place tan and sec live. Read backwards it says sec(x)² - 1 = tan(x)², and that form is the one that cancels most often.
Take (sec(x)² - 1)/tan(x). The top is tan(x)², so the fraction reads tan(x)²/tan(x). One tan x cancels and tan(x) is left, just as a²/a is a.
Challenge
Show that sec(x)² - tan(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 Angles3Pythagorean Identities
1 + tan²x = sec²x1 + cot²x = csc²xA Difference of SquaresSquaring a SumRecap: Three Forms