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

You may find it helpful to consider similar triangles, or look at Going round in circles.

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 \(0\)?

Take a look at the triangles in the diagram.

What similar triangles can you find?

What do you see as \(\delta \theta \rightarrow 0\)?

Diagram showing two right-angled triangles with angles theta + delta theta and two with angle theta

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 \(\delta \theta\)?

It may be helpful to label some points in the diagram.

How can the length of the hypotenuse of the red triangle be expressed in terms of \(\theta\)?

How can this help us to find the arc length?

Labelled diagram with right-angled triangle OQP with angle theta at 0, right-angle at Q, and triangle PST with right-angle at S and angle theta plus delta theta at P.
  • Which side length shows the increase in \(\tan \theta\) as \(\theta\) increases by \(\delta \theta\)?

\(QP\) is \(\tan \theta\) and \(QT\) is \(\tan (\theta+ \delta \theta)\).

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

When \(\delta \theta\) is small, one side of each blue triangle can be approximated by a circular arc.

As \(\delta \theta \rightarrow 0\), the red and blue triangles get closer and closer to being similar. Which are the corresponding sides in the red and blue triangles?