The formula: sin² θ + cos² θ = 1 is used to create the other two formulas.
When divided by cos² θ it creates:
When divided by sin² θ it creates:
Example: Simplify sin² θ + cos² θ - sec² θ using the Pythagorean Identities.
Since we know that sin² θ + cos² θ = 1, we can replace sin² θ + cos² θ of the equation with 1. Now the equation is: 1 - sec² θ.
Now we can go a step further. We can use the formula: tan² θ + 1 = sec² θ
However, you may be confused because the equation at hand and the formula looks different. Use the formula and MAKE it look like what we have. Basically, make tan² θ + 1 = sec² θ look like 1 - sec² θ.
In this case, subtract tan² θ and move it to the other side. tan² θ + 1 = sec² θ -tan² θ -tan² θ 1 = sec² θ -tan² θ
Now subtract sec² θ and move it to the other side! 1 = sec² θ - tan² θ -sec² θ-sec² θ 1 - sec² θ = -tan² θ
And voila! You can now substitute 1 - sec² θ with -tan² θ!! And that's the answer! Answer: -tan² θ
In case you were still confused, click on the button below to learn more about Pythagorean identities!