5.5 Formulae of sin (A±B), cos (A±B), tan (A±B), sin 2A, cos 2A, tan 2A

5.5 Formulae of sin (A ± B), cos (A ± B), tan (A ± B), sin 2A, cos 2A, tan 2A

(A) Compound Angles Formulae:
 sin(A±B)=sinAcosB±cosAsinB  cos(A±B)=cosAcosBsinAsinB  tan(A±B)= tanA±tanB 1tanAtanB

(B) Double Angle Formulae:
  • sin 2A = 2 sin A cos A
  • cos 2A = cos2 A – sin2 A
  • cos 2A = 2 cos2 A – 1
  • cos 2A = 1 – 2 sin2 A
  • tan2A= 2tanA 1 tan 2 A   

(C) Half Angle Formulae:
 sinA=2sin A 2 cos A 2  cosA= sin 2 A 2 cos 2 A 2    cosA=2 cos 2 A 2 1    cosA=12 cos 2 A 2  tanA= 2tan A 2 1 tan 2 A 2

5.5.1 Proving Trigonometric Identities using Addition Formula and Double Angle Formulae

Example 1:
Prove each of the following trigonometric identities.
(a)  sin( A+B )sin( AB ) cosAcosB =2tanB (b)  cos( A+B ) sinAcosB =cotAtanB (c) tan( A+ 45 o )= sinA+cosA cosAsinA

Solution:
(a)
LHS = sin( A+B )sin( AB ) cosAcosB = ( sinAcosB+cosAsinB )( sinAcosBcosAsinB ) cosAcosB = 2 cosA sinB cosA cosB = 2sinB cosB =2tanB=RHS (proven)
(b)
LHS = cos( A+B ) sinAcosB = cosAcosBsinAsinB sinAcosB = cosA cosB sinA cosB sinA sinB sinA cosB = cosA sinA sinB cosB =cotAtanB =RHS (proven) 
(c)
LHS =tan( A+ 45 o ) = tanA+tan 45 o 1tanAtan 45 o = tanA+1 1tanA tan 45 o =1 = sinA cosA +1 1 sinA cosA = sinA+cosA cosA × cosA cosAsinA = sinA+cosA cosAsinA =RHS (proven)