1sin (A + B)= sin A cos B + cos A sin B
Put B=A in the above formula
= Sin ( A + A ) = sin A cos A + cos A sin A
= 2 sin A cos A
∴ sin ( 2 A )= 2 sin A cos A
cos (A + B)= cos A cos B — sin A sinB
Put A=B
= cos (A + A)= cos A cos A — sin A sin A
= ( cos² A — sin² A ) ---------formula (1)
(sin² A + cos² A = 1, sin² A = 1— cos² A , cos² A = 1 — sin² A )-------------------------------(a)
= cos² A — ( 1— cos² A )
= cos² A — 1 + cos² A
= ( 2 cos² A — 1 )-------------- formula (2)
From (a) above put cos² A = 1 — sin² A in formula (1)
=( 1 — sin² A ) — sin² A
=( 1 — 2 sin² A )----------------formula (3)
∴ cos ( 2A ) = ( cos² A — sin² A ) , ( 2 cos² A — 1 ) & ( 1 — 2 sin² A )
sin ( 2A )= 2 sin A cos A
