The reciprocal trigonometric functions are:
When working with reciprocal functions, remember that there are asymptotes. They're undefined when their denominators equal zero!
Many students assume that $\sec x$ is the reciprocal of $\sin x$ and $\csc x$ is the reciprocal of $\cos x$. It's actually the other way round from what you'd expect, so $\sec x = \frac{1}{\cos x}$ and $\csc x = \frac{1}{\sin x}$.
Just like before, we can start with the Pythagorean theorem for a right-angled triangle:
$$o^2 + a^2 = h^2$$
Before, we divided both sides by $h^2$. However, we can also divide both sides by $o^2$ or $a^2$, and see what comes up.
Dividing both sides by $o^2$:
$$1 + \frac{a^2}{o^2} = \frac{h^2}{o^2}$$
But $\cot\theta = \frac{a}{o}$, and $\csc\theta = \frac{h}{o}$. So, we can say:
$$1 + \cot^2\theta = \csc^2\theta$$
Returning to the original Pythagorean theorem, we can also divide both sides by $a^2$ to get:
$$\frac{o^2}{a^2} + 1 = \frac{h^2}{a^2}$$
But $\tan\theta = \frac{o}a$, and $\sec\theta = \frac{h}a$. So:
$$\tan^2\theta + 1 = \sec^2\theta$$
Thus, to summarize the three Pythagorean trigonometric identities:
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