91Ó°ÊÓ

The tangent (\(\tan x\)) and cotangent (\(\cot x\)) functions are defined in terms of the basic trigonometric functions sine (\(\sin x\)) and cosine (\(\cos x\)) as follows: \[\tan x = \frac{\sin x}{\cos x}\] \[\cot x = \frac{\cos x}{\sin x}\]

The Quotient Rule is a method for finding the derivative of a function that is a quotient of two differentiable functions. If we have a function \(h(x) = \frac{f(x)}{g(x)}\), where both \(f(x)\) and \(g(x)\) are differentiable functions, then the derivative \(h'(x)\) is given by: \[h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}\]

We will now use the Quotient Rule to find the derivatives of \(\tan x\) and \(\cot x\). For \(\tan x\): We have \(f(x) = \sin x\) and \(g(x) = \cos x\). Their derivatives are \(f'(x) = \cos x\) and \(g'(x) = -\sin x\). Applying the Quotient Rule, we get: \[\frac{d}{dx}(\tan x) = \frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{(\cos x)^2} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x\] For \(\cot x\): We have \(f(x) = \cos x\) and \(g(x) = \sin x\). Their derivatives are \(f'(x) = -\sin x\) and \(g'(x) = \cos x\). Applying the Quotient Rule, we get: \[\frac{d}{dx}(\cot x) = \frac{-\sin x \cdot \sin x - \cos x \cdot \cos x}{(\sin x)^2} = \frac{-\sin^2 x - \cos^2 x}{\sin^2 x} = \frac{-1}{\sin^2 x} = -\csc^2 x\]

The Quotient Rule is used to determine the derivatives of \(\tan x\) and \(\cot x\) because these functions can be expressed as quotients of sine and cosine functions. With the Quotient Rule, we can then find their derivatives as \(\sec^2 x\) for \(\tan x\) and \(-\csc^2 x\) for \(\cot x\).