5.5.1 Proving Trigonometric Identities using Addition Formula and Double Angle Formulae (Examples)


Example 2:
Prove each of the following trigonometric identities.
(a)  1+cos2x sin2x =cotx (b) cotAsec2A=cotA+tan2A (c)  sinx 1cosx =cot x 2

Solution:
(a)
LHS = 1+cos2x sin2x = 1+( 2 cos 2 x1 ) 2sinxcosx = 2 cos 2 x 2 sinx cosx = cosx sinx =cotx=RHS (proven)
(b)
RHS =cotA+tan2A = cosA sinA + sin2A cos2A = cosAcos2A+sinAsin2A sinAcos2A = cosA( cos 2 A sin 2 A )+sinA( 2sinAcosA ) sinAcos2A = cos 3 AcosA sin 2 A+2 sin 2 AcosA sinAcos2A = cos 3 A+cosA sin 2 A sinAcos2A = cosA( cos 2 A+ sin 2 A ) sinAcos2A = cosA sinAcos2A sin 2 A+ cos 2 A=1 =( cosA sinA )( 1 cos2A ) =cotAsec2A
(c)
LHS = sinx 1cosx = 2sin x 2 cos x 2 1( 12si n 2 x 2 ) sinx=2sin x 2 cos x 2 , cosx=12 sin 2 x 2 = 2 sin x 2 cos x 2 2 si n 2 x 2 = cos x 2 sin x 2 =cot x 2 =RHS (proven)

Example 3:
(a) Given that sinP= 3 5  and sinQ= 5 13 ,  such that P is an acute angle and Q is an obtuse angle, without using tables or a calculator, find the value of cos (P + Q).
(b) Given that sinA= 3 5  and sinB= 12 13 ,  such that A and B are angles in the third and fourth quadrants respectively, without using tables or a calculator, find the value of sin (A  B).

Solution:
(a)
sinP= 3 5 ,       cosP= 4 5 sinQ= 5 13 ,     cosQ= 12 13 cos( P+Q ) =cosAcosBsinAsinB =( 4 5 )( 12 13 )( 3 5 )( 5 13 ) = 48 65 15 65 = 63 65

(b)

sinA= 3 5 ,       cosA= 4 5 sinB= 5 13 ,     cosB= 12 13 sin( AB ) =sinAcosBcosAsinB =( 3 5 )( 12 13 )( 4 5 )( 5 13 ) = 36 65 20 65 = 56 65