L2.4 C1 Proving Trigonometric Identities - Part 1
Completion requirements
Unit 2
Trigonometry
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Part 2.4C corresponds to section 6.3, starting on page 309 of your Pre-Calculus 12 textbook.
In 2.4A, you verified potential identities. A verification can be used to show an equation is not an identity, but it cannot guarantee an equation is an identity. To ensure an equation is an identity, a proof must be used.
A proof is a logical argument that unequivocally shows the truth of a statement. Proofs can take many forms, but this section focuses on a single type. To prove a trigonometric identity, you need to show the two sides of the equation are equivalent. You can do this by manipulating the equation, one side at a time, until the two sides are the same.
A proof is a logical argument that unequivocally shows the truth of a statement. Proofs can take many forms, but this section focuses on a single type. To prove a trigonometric identity, you need to show the two sides of the equation are equivalent. You can do this 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»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»tan«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«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».
The identity «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»tan«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»sin«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» can be used to prove this identity.
The left side equals the right side, so the identity has been proven.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»left«/mi»«mo»§#160;«/mo»«mi»side«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»tan«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mfenced»«mfrac»«mrow»«mi»sin«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mfenced»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»§#8226;«/mo»«mfrac»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«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»«mi»right«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»side«/mi»«mo»=«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«/mrow»«/mstyle»«/math»
The left side equals the right side, so the identity has been proven.