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

6.5 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±cosAsinBcos(A±B)=cosAcosBsinAsinBtan(A±B)=tanA±tanB1tanAtanB


(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=2tanA1tan2A   

(C) Half Angle Formulae:
   sinA=2sinA2cosA2   cosA=sin2A2cos2A2    cosA=2cos2A21  cosA=12cos2A2   tanA=2tanA21tan2A2


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+45o)=sinA+cosAcosAsinA

Solution:
(a)
LHS=sin(A+B)sin(AB)cosAcosB=(sinAcosB+cosAsinB)(sinAcosBcosAsinB)cosAcosB=2cosAsinBcosAcosB=2sinBcosB=2tanB=RHS (proven)



(b)
LHS=cos(A+B)sinAcosB=cosAcosBsinAsinBsinAcosB=cosAcosBsinAcosBsinAsinBsinAcosB=cosAsinAsinBcosB=cotAtanB=RHS(proven)  


(c)
LHS=tan(A+45o)=tanA+tan45o1tanAtan45o=tanA+11tanAtan45o=1=sinAcosA+11sinAcosA=sinA+cosAcosA×cosAcosAsinA=sinA+cosAcosAsinA=RHS(proven)

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