Expert: Paul Klarreich - 5/24/2008

Question
Prove the following identities
Cos2θ≡Cos^4θ-Sin^4θ
Cos3θ≡4Cos^3θ-3Cosθ
Sin3θ≡3Sinθ-4Cos^3θ
Cos4θ≡8Cos^4θ-8Cos^2θ+1
Sin4θ≡4(SinθCos^3θ-Sin^3θCosθ)

Answer
Questioner:   Henoc
Category:  Advanced Math
Private:  No
 
Subject:  Trig Identities
Question:  Prove the following identities
Cos2θ ≡ Cos^4θ-Sin^4θ
Cos3θ ≡ 4Cos^3θ-3Cosθ
Sin3θ ≡ 3Sinθ-4Cos^3θ
Cos4θ ≡ 8Cos^4θ-8Cos^2θ+1
Sin4θ ≡ 4(SinθCos^3θ-Sin^3θCosθ)
........................
Hi, Henoc,

Since you  didn't indicate what, if anything, you already did on these, I will assume you just don't know how to get started. (Sounds familiar)

Cos2θ ≡ Cos^4θ-Sin^4θ

Factor the right side.
..........................
Cos3θ ≡ 4Cos^3θ-3Cosθ

[I use t for theta.]

= cos(2t + t)
= cos(2t)cos t - sin(2t)sin t
= (     )cos t - 2sin t cos t sin t
Replace sin^2(t)
.........................
Sin3θ ≡ 3Sinθ-4Cos^3θ

Same stuff as before.
.........................
Cos4θ ≡ 8Cos^4θ-8Cos^2θ+1

= cos(2(2t))

= 2 cos^2(2t) - 1
........................
Sin4θ ≡ 4(SinθCos^3θ-Sin^3θCosθ)

= sin(2(2t))