Can you explain why the points given in the diagram below have coordinates \((1, \tan \theta)\) and \((\cot \theta, 1)\)?

Now move the slider for \(\delta \theta\).

  • What is the relationship between the blue angle and the red angle?

  • What can you say about the blue triangles and the red triangle as \(\delta \theta\) decreases to \(0\)?

Consider the blue triangle which has one vertex at \((1, \tan \theta)\).

  • What is the length of the circular arc drawn through \((1, \tan \theta)\) and subtended by \(\delta \theta\)?

  • Which side length shows the increase in \(\tan \theta\) as \(\theta\) increases by \(\delta \theta\)? This can be denoted by \(\delta(\tan\theta)\).

How can you use these ideas to find the derivative of \(\tan \theta\)?

What about the derivative of \(\sec \theta\)?

Can you take a similar approach to find the derivatives of \(\cot \theta\) and \(\cosec \theta\)? You may be able to do this in more than one way.