Solve tan(x)

Step-by-step solution

1. Rewrite $\tan x = \frac{\sin x}{\cos x}$.

2. Apply the quotient rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$.

3. With $f = \sin x$ ($f' = \cos x$) and $g = \cos x$ ($g' = -\sin x$): $\frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x}$.

4. Use the Pythagorean identity $\sin^2 + \cos^2 = 1$: $= \frac{1}{\cos^2 x} = \sec^2 x$.