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prove that: i. cosec2A - cot2A = tanA ii. 2sinAcos3A - 2sin3AcosA = sin4A/2
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
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
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
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