Identity from Math.StackExchange.com

Taken from my answer to “Proving a weird trig identity” at Math.StackExchange.com .

$$\Large{\frac{\cos\theta}{1-\sin\theta} = \frac{\sec\theta + \tan\theta}{1}}$$

In the diagram, $$\overline{AB}$$ is tangent to the unit circle at P. The “trig lengths” (except for $$|\overline{AQ}|$$) should be clear.

We note that $$\angle BPR \cong \angle RPP^\prime$$, since these inscribed angles subtend congruent arcs $$\stackrel{\frown}{PR}$$ and $$\stackrel{\frown}{RP^\prime}$$. Very little angle chasing gives that $$\triangle APQ$$ is isosceles, with $$\overline{AP}\cong \overline{AQ}$$ (justifying that last trig length). Then,
$$\triangle SPR \sim \triangle OQR \Longrightarrow \frac{|\overline{SP}|}{|\overline{SR}|}=\frac{|\overline{OQ}|}{|\overline{OR}|}\Longrightarrow \frac{\cos\theta}{1−\sin\theta}=\frac{\sec\theta+\tan\theta}{1}$$

Posted 14 May, 2015 by in Classroom, Trigonography