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

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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 ) = sin A cos B ± cos A sin B • cos ( A ± B ) = cos A cos B ∓ sin A sin B • tan ( A ± B ) = tan A ± tan B 1 ∓ tan A tan B


(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
  • tan 2 A = 2 tan A 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=1−2 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( A−B ) cosAcosB =2tanB (b)  cos( A+B ) sinAcosB =cotA−tanB (c) tan( A+ 45 o )= sinA+cosA cosA−sinA

Solution:
(a)
LHS = sin( A+B )−sin( A−B ) cosAcosB = ( sinAcosB+cosAsinB )−( sinAcosB−cosAsinB ) cosAcosB = 2 cosA sinB cosA cosB = 2sinB cosB =2tanB=RHS (proven)


(b)
L H S = cos ( A + B ) sin A cos B = cos A cos B − sin A sin B sin A cos B = cos A cos B sin A cos B − sin A sin B sin A cos B = cos A sin A − sin B cos B = cot A − tan B = R H S (proven)  


(c)
L H S = tan ( A + 45 o ) = t a n A + tan 45 o 1 − t a n A tan 45 o = t a n A + 1 1 − t a n A ← tan 45 o = 1 = sin A cos A + 1 1 − sin A cos A = sin A + cos A cos A × cos A cos A − sin A = sin A + cos A cos A − sin A = R H S (proven)