Prove the Pythagorean Identity sin^2(θ) + cos^2(θ) = 1
There are three Pythagorean Identities, and you can get from one to the other with a little algebra magic. One of these identities is: sin^2(θ) + cos^2(θ) = 1. But how do we know that adding “sine of theta squared” plus “cosine of theta squared” truly equals one? (Fyi, sin^2(θ) is the same thing as sin(θ)^2).
To write a trigonometric proof, we need the help of our Trig Toolbox. These are equations that we use all the time in trigonometry, and if you know them well, you can use them to solve tons of problems. So let’s look in the toolbox and see if we can pick out any of the equations in there that, with a bit of algebra, can be turned into sin^2(θ) + cos^2(θ) = 1.
Right off the bat, the definitions of sin and cos look like they will come in handy. If we square them, they will look closer to the way they look in the Pythagorean Identity we are trying to prove.
So now we have some squared opposites, adjacents, and hypotenuses. Is there anything in our toolbox with squares of those? Yes, the Pythagorean Theorem!
How can we make the Pythagorean theorem resemble the terms we have in the dashed green and orange boxes? We need the opposite^2 to be divided by the hypotenuse^2, we need the adjacent^2 to be divided by the hypotenuse^2, and we need a 1 on the other side of the equation. Well luckily for us that can all be accomplished in one step! We need to divide BOTH sides of the equation by hypotenuse^2. (Fyi, dividing a variable by itself results in the value 1.)
Almost done! Now we just need to do some substitution. We know from our first step that opposite^2/hypotenuse^2 = sin^2(θ), so we can put sin^2(θ) in its place. We also know from our first step that adjacent^2/hypotenuse^2 = cos^2(θ), so we can put cos^2(θ) in its place. And we know that dividing a variable by itself (hypotenuse^2/hypotenuse^2) equals 1, so we can put a 1 in its place.
And we’re done!
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