Consider the functions \(f(x)=\tan x\) and \(g(x)=\cot x\).
- Explain why \(\cot x \neq \dfrac{1}{\tan x}\) for all values of \(x\). (2 marks)
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- On the same set of axes below, sketch \(y=\tan x\) and \(y=\cot x\) for \(0<x<\pi\), identifying any points where the graphs intersect. (2 marks)
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a. \(\text{At}\ \ x=\dfrac{\pi}{2}:\)
\(\cot \dfrac{\pi}{2}=\dfrac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}}=\dfrac{0}{1}=0 \ \Rightarrow \ \text{defined}\).
\(\tan \dfrac{\pi}{2}=\dfrac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}}=\dfrac{1}{0} \Rightarrow \ \text{undefined}\).
\(\dfrac{1}{\tan \frac{\pi}{2}}\ \ \text{is therefore undefined}\).
\(\therefore \cot x \neq \dfrac{1}{\tan x} \ \ \text{for all values of }\ x\).
b.
a. \(\text{At}\ \ x=\dfrac{\pi}{2}:\)
\(\cot \dfrac{\pi}{2}=\dfrac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}}=\dfrac{0}{1}=0 \ \Rightarrow \ \text{defined}\).
\(\tan \dfrac{\pi}{2}=\dfrac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}}=\dfrac{1}{0} \Rightarrow \ \text{undefined}\).
\(\dfrac{1}{\tan \frac{\pi}{2}}\ \ \text{is therefore undefined}\).
\(\therefore \cot x \neq \dfrac{1}{\tan x} \ \ \text{for all values of }\ x\).
b.