Building blocks

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Problem

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.