Theorem sin2t

Description: Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008)

	Ref	Expression
Assertion	sin2t	$$⊢A\in ℂ\to \sin 2A=2\sin A\cos A$$

Proof

Step	Hyp	Ref	Expression
1		2times	$$⊢A\in ℂ\to 2A=A+A$$
2	1	fveq2d	$$⊢A\in ℂ\to \sin 2A=\sin \left(A+A\right)$$
3		coscl	$$⊢A\in ℂ\to \cos A\in ℂ$$
4		sincl	$$⊢A\in ℂ\to \sin A\in ℂ$$
5	3 4	mulcomd	$$⊢A\in ℂ\to \cos A\sin A=\sin A\cos A$$
6	5	oveq2d	$$⊢A\in ℂ\to \sin A\cos A+\cos A\sin A=\sin A\cos A+\sin A\cos A$$
7		sinadd	$$⊢\left(A\in ℂ\wedge A\in ℂ\right)\to \sin \left(A+A\right)=\sin A\cos A+\cos A\sin A$$
8	7	anidms	$$⊢A\in ℂ\to \sin \left(A+A\right)=\sin A\cos A+\cos A\sin A$$
9	4 3	mulcld	$$⊢A\in ℂ\to \sin A\cos A\in ℂ$$
10	9	2timesd	$$⊢A\in ℂ\to 2\sin A\cos A=\sin A\cos A+\sin A\cos A$$
11	6 8 10	3eqtr4d	$$⊢A\in ℂ\to \sin \left(A+A\right)=2\sin A\cos A$$
12	2 11	eqtrd	$$⊢A\in ℂ\to \sin 2A=2\sin A\cos A$$