sin²θ + cos²θ = 1 for every angle. It is the single most used identity in trigonometry.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Where it comes from
The terminal point (cos θ, sin θ) lies on a circle of radius 1, so x² + y² = 1. The identity is that equation renamed.
What it does
Given one ratio, it produces the other up to sign. The quadrant then settles the sign.
Two rearrangements
Dividing by cos²θ gives tan²θ + 1 = sec²θ. Dividing by sin²θ gives 1 + cot²θ = csc²θ.
Derive rather than memorise
Both rearrangements come from one division. Remembering the original and the two divisions is less to carry.
In proofs
It is what lets a squared expression be replaced, which is how most identity proofs make progress.
Pythagoras on a circle of radius one
A point on the unit circle is (cos θ, sin θ) at distance 1 from the origin, so cos²θ + sin²θ = 1. The identity is the distance formula applied to the circle's defining property.
Two more follow by division
Dividing through by cos²θ gives 1 + tan²θ = sec²θ; by sin²θ gives cot²θ + 1 = csc²θ. Only one needs memorising, since the others are one step away.
What sin²θ means
It is (sin θ)², not sin(θ²). The exponent sits between the function name and its argument by convention, which is unusual notation and worth reading carefully.
Its use is conversion
It lets sin²θ be replaced by 1 − cos²θ, turning a mixed expression into one written in a single function. That conversion is the standard first move when simplifying.
Step 2: Try It Yourself
Tap and try it out.
- Angle55° = π/3 rad
- x-coordinate0.574
- y-coordinate0.819
- cos 55°0.574
- sin 55°0.819
The cosine is the x-coordinate and the sine is the y-coordinate. They are not merely equal — they are the same two numbers, which is why angles past 90° cause no trouble.
Step 3: Watch an Example
One step at a time.
Watch Marcus Find a Missing Ratio
Marcus knows sin θ = 0.6 and that θ is acute.
- Step 1
The identity gives cos²θ = 1 − sin²θ.
Step 4: Your Turn
Practice makes it stick.
The Square
Problem 1 of 2
sin²θ = 0.36. What is cos²θ?
The Ratio
Problem 2 of 2
sin θ = 0.6 with θ acute. What is cos θ?
Use the Identity
1 of 8
sin²θ = 0.49. What is cos²θ?
2 of 8
cos θ = 0.8 with θ acute. What is sin θ?
3 of 8
What is sin²θ + cos²θ for any angle?
4 of 8
tan²θ + 1 = 9. What is sec²θ?
5 of 8
cot²θ = 15. What is csc²θ?
6 of 8
sec²θ − tan²θ equals what number?
7 of 8
Match each expression with its simplest form.
Tap a card on the left to start.
8 of 8
sin²θ = 0.25. What is cos²θ?
Step 5: Quick Check
Show what you know.
Question 1 of 2
sin²θ = 0.64. What is cos²θ?
Question 2 of 2
Where does the Pythagorean identity come from?
What You Learned
- sin²θ + cos²θ = 1 holds for every angle.
- It is the unit circle equation with the coordinates renamed.
- Dividing by cos²θ or sin²θ gives the other two forms.