TIE - The Pythagorean Identity Lesson

The Pythagorean Identity

In Geometry, you learned a very important theorem known as the Pythagorean Theorem. Now, we are going to apply the Pythagorean Theorem to verify some Trigonometry identities. So let's think about this in terms of sine and cosine.

If we let: $x^2+y^2=1$

Now, divide both sides by $r^2$: $\frac{x^2+y^2}{r^2}=\frac{r^2}{r^2}$

And re-arranged a bit: $\frac{x^2}{r^2}+\frac{y^2}{r^2}=1$

Apply exponent rules: $\left(\frac{x}{r}\right)^2+\left(\frac{y}{r}\right)^2=1$

Replace with sine and cosine: $\left(\sin\theta\right)^2+\left(\cos\theta\right)^2=1,\:$  which can also be written this way: $\sin^2\theta+\cos^2\theta=1.$

This result is called a Pythagorean Identity. It can be written in different forms by manipulating the original equation. For example,

                                      $\sin^2\theta+\cos^2\theta=1$ 

                                      $\cos^2\theta=1-\sin^2\theta$

                                      $\sin^2\theta=1-\cos^2\theta$

So now, we can use the Pythagorean Identity to verify identity problems.

On a separate sheet of paper, verify the identities below:

$1.\:\left(\cos^2\theta-1\right)\left(\tan^2\theta+1\right)=-\tan^2\theta \\ 2.\:\frac{\left(\tan^2\theta+1\right)}{\tan^2\theta}=1+\cot^2\theta \\ 3.\:\tan B+\cot B=\csc B\sec B$

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