Trigonometric formula sin(2x)=2 sin x cos x

Saturday 18 February 2023, by Nadir Soualem

All the versions of this article: <English> <français>

We will show that for any real element x the trigonometric formula sin(2x)=2 sin x cos xx

Let’s show that:

$$\forall x \in \mathbb{R}, \quad \sin(2x)=2 \sin x \cos x$$

Proof/Demonstration

We will use the addition formula previously demonstrated:

$$\forall a,b \in \mathbb{R}, \quad \sin (a+b)=\sin a \cos b + \cos a \sin b$$

Taking $a=b=x$. We have $\forall x \in \mathbb{R}$:

$$ \begin{aligned} \sin(2x)= \sin (x+x) &=\sin x \cos x + \cos x \sin x\\ & =\sin x \cos x + \sin x \cos x \\ &= 2 \sin x \cos x \end{aligned} $$

Conclusion

$$\forall x in \mathbb{R}, \quad \sin(2x)=2 \sin x \cos x$$