TRIPLE ANGLE IDENTITIES

Using the following double angle identities, we can derive triple angle identities.

sin2A = 2sinAcosA

cos2A = 2cos²A - 1

cos2A = 1 - 2sin²A

tan2A = 2tanA/(1 - tan²A)

Identity 1 : sin3A = 3sinA - 4sin³A

Proof : 

sin3A = sin(2A + A)

= sin2AcosA + cos2AsinA

= 2sinAcosAcosA + (1 - 2sin²A)sinA

= 2sinAcos²A + sinA - 2sin³A

= 2sinA(1 - sin²A) + sinA - 2sin³A

= 2sinA - 2sin³A + sinA - 2sin³A

= 3sinA - 4sin³A

Identity 2 : cos3A = 4cos³A - 3cosA

Proof : 

cos3A = cos(2A + A)

= cos2AcosA - sin2AsinA

= (2cos²A - 1)cosA - 2sinAcosAsinA

= 2cos³A - cosA - 2cosAsin²A

= 2cos³A - cosA - 2cosA(1 - cos²A)

= 2cos³A - cosA - 2cosA + 2cos³A

= 4cos³A - 3cosA

Identity 3 : tan3A = (3tanA - tan³A)/(1 - 3tan²A)

Proof : 

tan3A = tan(2A + A)

= (tan2A + tanA)/(1 - tan2AtanA)

Using double angle identity tan2A = 2tanA/(1 - tan²A), 

= (3tanA - tan³A)/(1 - 3tan²A)

Double and Triple Angle Identities

Sine : 

sin2A = 2sinAcosA

sin2A = 2tanA/(1 + tan²A)

sin3A = 3sinA - 4sin³A

Cosine : 

cos2A = cos²A - sin²A

cos2A = 2cos²A - 1

cos2A = 1 - 2sin²A

cos2A = (1 - tan²A)/(1 + tan²A)

cos3A = 4cos³A - 3cosA

Tangent : 

tan2A = 2tanA/(1 - tan²A)

tan3A = (3tanA - tan³A)/(1 - 3tan²A)

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