Question

What limit facts and trigonometric identities are used in

the proof that$\frac{d}{dx}\left(\sin x\right)=\cos x?$

Short answer

limit facts $\underset{h\to 0}{\lim }\frac{(\cos h-1)}{h}=0\&\underset{h\to 0}{\lim }\frac{\sin h}{h}=1$and sum identity for the sine $\sin (a+b)=\sin a\cos b+\cos a\sin b$are used in the proof of$\frac{d}{dx}\left(\sin x\right)=\cos x.$

Step-by-step solution

Step 1. Given information. 

the given formula is$\frac{d}{dx}\left(\sin x\right)=\cos x.$

Step 2. proof.

Use the definition of the derivative to derivate $f\left(x\right)=\sin x.$

$\frac{d}{dx}\sin x=\underset{h\to 0}{\lim }\frac{\sin (x+h)-\sin x}{h}$

Apply sum identity for the sine ,$\sin (a+b)=\sin a\cos b+\cos a\sin b.$

role="math" localid="1648766873735" $$\begin{gathered} \frac{d}{dx}\sin x=\underset{h\to 0}{\lim }\frac{\sin x\cos h+\cos x\sin h-\sin x}{h} \\ \frac{d}{dx}\sin x=\underset{h\to 0}{\lim }\frac{\sin x(\cos h-1)+\cos x\sin h}{h} \\ \frac{d}{dx}\sin x=\sin x\left(\underset{h\to 0}{\lim }\frac{(\cos h-1)}{h}\right)+\cos x\left(\underset{h\to 0}{\lim }\frac{\sin h}{h}\right) \end{gathered}$$

Use limit fact $\underset{h\to 0}{\lim }\frac{(\cos h-1)}{h}=0\&\underset{h\to 0}{\lim }\frac{\sin h}{h}=1$

$$\begin{gathered} \frac{d}{dx}\sin x=\sin x\left(0\right)+\cos x\left(1\right) \\ \frac{d}{dx}\sin x=\cos x \end{gathered}$$