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Home > Mathematics > Trigonometry > Trigonometric formula sin(2x)=2 sin x cos x
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We will show that for any real element x the trigonometric formula sin(2x)=2 sin x cos xx
Let’s show that:
$$\forall x \in \mathbb{R}, \quad \sin(2x)=2 \sin x \cos x$$
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
$$\forall x in \mathbb{R}, \quad \sin(2x)=2 \sin x \cos x$$