The Story Of sin^2(θ) + cos^2(θ) = 1

Have you ever wondered why does this spooky... no not spooky rather a simple and pretty trigonometry identity "sin^2(θ) + cos^2(θ) = 1" came into existence  ? You may probably remember this identity from your high school mathematics. But if you don't then there is no problem. There is no need for you to sweat over it. Just simply google this and you will find about it. Today we will see the story of this identity.

But before diving into the story of sin^2(θ) + cos^2(θ) = 1, we need to travel back into time to ancient Ionian Greek so that we can meet Mr. Pythagoras aka Pythagoras of Samos and get to know about his famous Pythagoras Theorem. So, ladies and gentleman tighten your seat belt as our time machine is going back into time. Close your eyes and enjoy the journey. Three...Two...One... Cool we have finally reached to our destination. Mr. Pythagoras is telling his theorem so read carefully else you may cube a side. 

Great! We got to know about Pythagoras Theorem. Now let's go back to our present time. Three...Two...One... You can open your eyes as we have reached in the year 2021. 

Well you may be wondering why do we need Mr. Pythagoras's theorem for reading the story of our simple yet beautiful identity. It turns out that the above theorem is an important component of our story. It is the base that makes our identity to exist.

The Unit Circle In The Play

Let us take a unit circle. A unit circle has following features:

1. It is a circle. (That's obvious 😆)
2. It has a radius of one unit.
3. It is centered at the origin usually denoted by English alphabet O.

The Entry Of Point P

Now we will take some arbitrary point say P on the circumference of the unit circle having coordinates a (x coordinate) and b (y coordinate). Join O and P thus forming radius OP = 1 unit. Draw perpendicular PM to x-axis, intersecting x-axis at the point M. We can see that the perpendicular distance PM is equal to the y coordinate 'b' of the point P and the distance OM is equal to the x coordinate 'a' of the point P.

From the above figure it can be concluded that triangle POM is a right-angled triangle where ∠PMO = 90°.  Let us define an angle say POM = θ. We will take some concepts of trigonometry to reach to the main part of our story.

We know that sin(θ) = opposite/hypotenuse and cos(θ) = adjacent/hypotenuse. 

So in our triangle POM:

PM = b

OM = a

OP = 1

sin(θ) = PM/OP = b/1 = b   ....(1)

cos(θ) = OM/OP = a/1 = a  ...(2)

From (1) and (2) it can be concluded that the 'x coordinate' of point P is also 'cos(θ)' and the 'y coordinate' of point P is also 'sin(θ)' .

The Last Part

This is the main and the last part of our story. We know that in any right-angled triangle we have 

                             P^2 + B^2 = H^2   (All thanks to Mr. Pythagoras)

                        where P = Perpendicular, B = Base and H = Hypotenuse

Going back to our right-angled triangle POM in the unit circle we have:

                            

                            PM^2 + OM^2 = OP^2  ...(3)

But we know that PM is 'b' which in turn is 'sin(θ)' from (1) and similarly OM is 'a' which in turn is 'cos(θ)' from (2)

So (3) becomes:

                                sin^2(θ) + cos^2(θ) = 1

Ta-da! We did it!

Isn't it simple yet beautiful at the same time? We finally found out the story of sin^2(θ) + cos^2(θ) = 1. Just by using Unit Circle, Trigonometry and Pythagoras Theorem we simply found out a trigonometry identity. 

You see everything in mathematics is connected. You only need to look for that connection and eventually you will find connection with the universe :)

Happy Learning!

For practice you can try to prove why

1.  1 + tan^2(θ) = sec^2(θ)
2.  1 + cot^2(θ) = csc^2(θ)

You can take the help from the principles discussed above!