Unit 4A

Trigonometry Part 1

Lesson 2: Trigonometric Identities

Proving an Identity

A proof is a logical argument that unequivocally shows the truth of a statement. Proofs can take many forms, but for this Lesson, showing the two sides of an equation are equivalent is sufficient for proving an equation is an identity. This can be done by manipulating the equation, one side at a time, until the two sides are the same.

Prove the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»cot«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math» is an identity for all permissible values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».


Left Side
Right Side
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨center left¨»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»cot«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mfenced»«mstyle displaystyle=¨true¨»«mfrac»«mi»cosx«/mi»«mi»sinx«/mi»«/mfrac»«/mstyle»«/mfenced»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»cos«/mi»«mi»x«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»sin«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»sin«/mi»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» 
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math» 
 
Steps to simplify this expression:

  1. Replace «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cot«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math» with a quotient identity.
  2. Divide the fractions by multiplying the expression in the numerator by the reciprocal of the expression in the denominator.
  3. Reduce like terms.

The left side equals the right side, so the identity has been proven.