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would in algebra by multiplying by LCD . Factorise where possible and simplify. • End your proof with LHS = RHS. Nete: Identities involving compound angles do not necessarily follow the steps above. WORKED EXAMPLES 1 Prove that 1- cos 2x = tan x sin 2x. Proof: LHS = 1 - cos 2x = 1- (1 - 2 sin2 x) = 2 sin? x RHS = tan x sin 2x sin x COS X x 2 sin x.cos x = 2 sin? x . LHS = RHS cos 2x 1 + sin 2x cOs x- sin x COs x + sin x Prove that Proof: cos² x – sin2 x COs 2x 1+ sin 2x LHS = (sin? x + cos? x) + 2 sin x cos x (cos x - sin x)(cos x + sin x) (cos x + sin x)(cos x + sin x) | Difference of two squares | Perfect square trinomial %3D (cos x - sin x) (cos x + sin x) = RHS %3D 1- cos 2x 1 + cos 2x 3. Prove that tan?x = Proof: 1- cos 2x 1 + cos 2x RHS = 1-(1- 2 sin2 x) 1 + 2 cos?x - 1 2 sin2 x 2 cos? x | Use the double angle identities that cancel the 1. = tan?x = LHS

would in algebra by multiplying by LCD . Factorise where possible and simplify. • End your proof with LHS = RHS. Nete: Identities involving compound angles do not necessarily follow the steps above. WORKED EXAMPLES 1 Prove that 1- cos 2x = tan x sin 2x. Proof: LHS = 1 - cos 2x = 1- (1 - 2 sin2 x) = 2 sin? x RHS = tan x sin 2x sin x COS X x 2 sin x.cos x = 2 sin? x . LHS = RHS cos 2x 1 + sin 2x cOs x- sin x COs x + sin x Prove that Proof: cos² x – sin2 x COs 2x 1+ sin 2x LHS = (sin? x + cos? x) + 2 sin x cos x (cos x - sin x)(cos x + sin x) (cos x + sin x)(cos x + sin x) | Difference of two squares | Perfect square trinomial %3D (cos x - sin x) (cos x + sin x) = RHS %3D 1- cos 2x 1 + cos 2x 3. Prove that tan?x = Proof: 1- cos 2x 1 + cos 2x RHS = 1-(1- 2 sin2 x) 1 + 2 cos?x - 1 2 sin2 x 2 cos? x | Use the double angle identities that cancel the 1. = tan?x = LHS

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter5: Identities And Formulas
Section: Chapter Questions
Problem 2GP
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mplify as you would in algebra by multiplying LCD by LCD . Factorise where possible and simplify. • End your proof with LHS = RHS. Note: Identities involving compound angles do not necessarily follow the steps above. WORKED EXAMPLES Prove that 1- cos 2x = tan x sin 2x. Proof: Doub sin 26 LHS = 1 - cos 2x RHS = tan x sin 2x cos 2 = 1- (1 - 2 sin2 x) sin x COS X cos 2 x 2 sin x.cos X = 2 sin? x cos 2 = 2 sin? x .. LHS = RHS cos 2x 1 + sin 2x Cos X - sin x COS X + sin x Prove that Proof: cos? x – sin? x cos 2x 1+ sin 2x Othe LHS = (sin? x + cos x) + 2 sin x cos x tan 0 sin? (cos x – sin x)(cos x + sin x) (cos x + sin x)(cos x + sin x) | Difference of two squares Perfect square trinomial cos sin? (cos x - sin x) = RHS (cos x + sin x) 1- cos 2x 1 + cos 2x Prove that tan²x %3D Proof: 1- cos 2x 1 + cos 2x RHS = 1- (1 – 2 sin² x) 1 + 2 cos?x – 1 2 sin2 x 2 cos2 x | Use the double angle identities that cancel the 1. %3D = tan2x = LHS Unit 3 Prove identities using compound and do
Transcribed Image Text:mplify as you would in algebra by multiplying LCD by LCD . Factorise where possible and simplify. • End your proof with LHS = RHS. Note: Identities involving compound angles do not necessarily follow the steps above. WORKED EXAMPLES Prove that 1- cos 2x = tan x sin 2x. Proof: Doub sin 26 LHS = 1 - cos 2x RHS = tan x sin 2x cos 2 = 1- (1 - 2 sin2 x) sin x COS X cos 2 x 2 sin x.cos X = 2 sin? x cos 2 = 2 sin? x .. LHS = RHS cos 2x 1 + sin 2x Cos X - sin x COS X + sin x Prove that Proof: cos? x – sin? x cos 2x 1+ sin 2x Othe LHS = (sin? x + cos x) + 2 sin x cos x tan 0 sin? (cos x – sin x)(cos x + sin x) (cos x + sin x)(cos x + sin x) | Difference of two squares Perfect square trinomial cos sin? (cos x - sin x) = RHS (cos x + sin x) 1- cos 2x 1 + cos 2x Prove that tan²x %3D Proof: 1- cos 2x 1 + cos 2x RHS = 1- (1 – 2 sin² x) 1 + 2 cos?x – 1 2 sin2 x 2 cos2 x | Use the double angle identities that cancel the 1. %3D = tan2x = LHS Unit 3 Prove identities using compound and do
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