prove that:

i. cosec2A - cot2A = tanA

ii. 2sinAcos³A - 2sin³AcosA = sin4A/2

1) COSEC2A - COT2A = TANA

 LHS= 1/SIN2A - COS2A/SIN2A

      =1-COS2A/SIN2A

COS2A = 1-2SIN²A       &           sin2a =2SINACOSA  ,   BY USING THESE

 

 LHS=  [1- (1-2SIN²A)]/2SINACOSA

      =2SIN²A/2SINACOSA=SINA/COSA

      =TANA = RHS

HENCE PROVED

 

2)

   2SINACOS³A - 2SIN³ACOSA = SIN4A/2

 LHS =2SINACOSA(COS²A - SIN²A)

 COS²A - SIN²A =COS2A   &   2SINACOSA =SIN2A

 LHS =SIN2ACOS2A

        =(2SIN2ACOS2A)/2

        =SIN4A/2                                  (using sin2x = 2sinxcosx)

hence proved

Cosec2A= cot2A+tanA=> RHS=cot2A+tanACos2A/sin2A+sinA/cosACos2A*cosA+sin2A*sinA/sin2A.cosaCos(2A-A)/sin2A.cosACosA/sin2A.cosA1/sin2A=cosec2A