tan of a Half Angle
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 41 of 52.
Divide the half-angle sine by the half-angle cosine and tidy up, and the half-angle tangent needs no square root at all:
tan(x/2) = (1 - cos(x))/sin(x)
tan(x/2) = sin(x)/(1 + cos(x))
Both hold wherever both sides are defined. With x = 2A they turn a double angle back into A: sin(2A)/(1 + cos(2A)) is tan(A).
At 90°: (1 - cos(90°))/sin(90°) = 1, and tan 45° is indeed 1.
Challenge
Prove that csc(x) - cot(x) 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 Angles