Pythagorean Identities
The Pythagorean theorem is one of the most famous results in mathematics. But did you know it hides inside every trigonometric function? In this lesson you will discover the three Pythagorean identities that connect sine, cosine, tangent, and their reciprocals — identities so powerful they appear in almost every trigonometry problem you will ever solve.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
You already know that $\sin^2 \theta + \cos^2 \theta = 1$. What do you think happens if you divide every term in this equation by $\sin^2 \theta$? And what happens if you divide every term by $\cos^2 \theta$? Try to predict the results before reading on.
One simple identity, two divisions, three powerful results. The second and third identities are derived by dividing the first identity by $\cos^2 \theta$ and $\sin^2 \theta$ respectively. All three are true for every value of $\theta$ where the functions are defined.
The fundamental identity $\sin^2 \theta + \cos^2 \theta = 1$ comes directly from the unit circle equation $x^2 + y^2 = 1$. Dividing by $\cos^2 \theta$ gives the tangent-secant identity. Dividing by $\sin^2 \theta$ gives the cotangent-cosecant identity.
Fundamental identity: $\sin^2 \theta + \cos^2 \theta = 1$ (from the unit circle $x^2 + y^2 = 1$); Divide by $\cos^2 \theta$: $1 + \tan^2 \theta = \sec^2 \theta$ (provided $\cos \theta \neq 0$)
Pause — copy the fundamental identity $\sin^2\theta + \cos^2\theta = 1$ and the derived identity $1 + \tan^2\theta = \sec^2\theta$ (obtained by dividing through by $\cos^2\theta$) into your book.
True or false: The identity $1 + \tan^2 \theta = \sec^2 \theta$ is derived by dividing $\sin^2 \theta + \cos^2 \theta = 1$ by $\sin^2 \theta$.
Key facts
- The three Pythagorean identities
- How to derive the tangent-secant and cotangent-cosecant identities
- Common rearrangements of each identity
Concepts
- Why all three identities come from the unit circle
- How division transforms one identity into another
- When each identity is most useful
Skills
- Prove the three Pythagorean identities from first principles
- Use the identities to find missing trig values
- Simplify trigonometric expressions using identities
We just saw that dividing $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$ gives $\tan^2\theta + 1 = \sec^2\theta$. That raises a question: where do these identities actually come from — and how do we use them to rearrange into a form we need? This card answers it → the derivation starts from the unit circle equation $x^2 + y^2 = 1$ and the rearrangements are shown step by step.
The fundamental identity comes directly from the unit circle equation $x^2 + y^2 = 1$, where $x = \cos \theta$ and $y = \sin \theta$.
Deriving Identity 2: Divide every term by $\cos^2 \theta$ (provided $\cos \theta \neq 0$):
Deriving Identity 3: Divide every term by $\sin^2 \theta$ (provided $\sin \theta \neq 0$):
Each identity is useful in different situations. Memorise the rearrangements too:
- $\sin^2 \theta = 1 - \cos^2 \theta$ · $\cos^2 \theta = 1 - \sin^2 \theta$
- $\tan^2 \theta = \sec^2 \theta - 1$ · $\sec^2 \theta - \tan^2 \theta = 1$
- $\cot^2 \theta = \csc^2 \theta - 1$ · $\csc^2 \theta - \cot^2 \theta = 1$
Start the derivation with the unit circle: at angle $\theta$, the point is $(\cos \theta, \sin \theta)$, and $x^2 + y^2 = 1$; Dividing by $\cos^2 \theta$ gives $\tan^2 \theta + 1 = \sec^2 \theta$ — tangent always pairs with...
Pause — copy the unit-circle derivation pathway: $(x^2 + y^2 = 1) \to (\sin^2\theta + \cos^2\theta = 1)$, then divide by $\cos^2\theta$ to get $\tan^2\theta + 1 = \sec^2\theta$ into your book.
Quick check: Which identity do you obtain by dividing $\sin^2 \theta + \cos^2 \theta = 1$ by $\cos^2 \theta$?
We just saw the derivation: dividing $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$ yields $\tan^2\theta + 1 = \sec^2\theta$. That raises a question: how do you write this as a formal proof — starting from one side and reaching the other? This card answers it → begin with the fundamental identity, divide by $\cos^2\theta$, simplify, and always state the restriction $\cos\theta \neq 0$.
Prove that $1 + \tan^2 \theta = \sec^2 \theta$.
Proof structure: start from $\sin^2 \theta + \cos^2 \theta = 1$, divide through, simplify; Always state the restriction: "provided $\cos \theta \neq 0$"
Pause — copy the proof structure (start from the fundamental identity, divide, simplify) and the requirement to state "provided $\cos\theta \neq 0$" into your book.
Fill the blanks: drag each token into the matching blank to complete the proof step.
Dividing $\sin^2 \theta + \cos^2 \theta = 1$ by ___ gives ___ $+ 1 =$ ___.
We just saw how to prove an identity by manipulating one side to match the other. That raises a question: identities are also useful backwards — if I know $\sin\theta$, can I find $\cos\theta$ without a calculator? This card answers it → yes: rearrange $\sin^2\theta + \cos^2\theta = 1$ and substitute the known value.
If $\sec \theta = \frac{5}{4}$ and $\theta$ is acute, find $\tan \theta$.
Identify which identity links the known and unknown trig functions; Substitute the given value, isolate the unknown squared term, then take the square root
Pause — copy the three-step procedure: (1) identify which identity links known to unknown, (2) substitute, (3) isolate the squared term then take the square root into your book.
Quick check: If $\tan \theta = 2$ and $\theta$ is acute, which identity gives $\sec \theta$?
We just saw how to use a Pythagorean identity to find a missing trig value from a known one. That raises a question: what if the expression contains $\sec^2\theta - 1$ — does that simplify to something neater? This card answers it → yes: $\sec^2\theta - 1 = \tan^2\theta$ is a rearrangement of Identity 2; spotting this pattern is the key skill.
Simplify $\frac{\sec^2 \theta - 1}{\tan \theta}$.
Simplification strategy: identify a Pythagorean identity "lurking" in the numerator or denominator; $\sec^2 \theta - 1 = \tan^2 \theta$ is the rearrangement of Identity 2 — recognise it instantly
Pause — copy the simplification strategy (look for a Pythagorean identity lurking in the expression) and the rearrangement $\sec^2\theta - 1 = \tan^2\theta$ into your book.
Odd one out: Three of these expressions simplify to $1$ using Pythagorean identities. Which one does NOT?
Students sometimes write $1 + \tan^2 \theta = \csc^2 \theta$. The identity with tangent always pairs with secant, not cosecant. Remember the pairs: sine-cosine, tangent-secant, cotangent-cosecant.
Some students write $\tan^2 \theta = \sec^2 \theta$ without the $+1$. Memorise the complete equation: $1 + \tan^2 \theta = \sec^2 \theta$.
When you divide by $\cos^2 \theta$, you are assuming $\cos \theta \neq 0$. In formal proofs, this restriction should be mentioned. Always note: "provided $\cos \theta \neq 0$" when dividing by $\cos^2 \theta$.
Use the appropriate identity to find each exact missing value.
If $\tan \theta = 2$ and $\theta$ is acute, find $\sec \theta$.
If $\csc \theta = 3$ and $\frac{\pi}{2} < \theta < \pi$, find $\cot \theta$.
If $\cos \theta = -\frac{2}{3}$ and $\pi < \theta < \frac{3\pi}{2}$, find $\sin \theta$.
If $\sec \theta = -\frac{5}{3}$ and $\frac{\pi}{2} < \theta < \pi$, find $\tan \theta$.
Simplify $\sin^2 \theta(1 + \cot^2 \theta)$.
Return to your original answer from Section 01. Dividing $\sin^2 \theta + \cos^2 \theta = 1$ by $\cos^2 \theta$:
Dividing by $\sin^2 \theta$:
One simple identity, two divisions, three powerful results. This is the elegance of the Pythagorean identities.
Pick your answer, then rate your confidence — that tells the system what to drill next.
Prove the cotangent-cosecant identity
Prove that $1 + \cot^2 \theta = \csc^2 \theta$ starting from $\sin^2 \theta + \cos^2 \theta = 1$. State any restrictions. (3 marks)
View comprehensive answer
Proof:
Start with $\sin^2 \theta + \cos^2 \theta = 1$.
Divide every term by $\sin^2 \theta$ (provided $\sin \theta \neq 0$):
Simplify:
Restriction: $\sin \theta \neq 0$, i.e. $\theta \neq n\pi$ for $n \in \mathbb{Z}$.
Finding sin and cos from tangent
If $\tan \theta = -\frac{4}{3}$ and $\frac{3\pi}{2} < \theta < 2\pi$, find the exact values of $\sin \theta$ and $\cos \theta$. Show all working. (3 marks)
View comprehensive answer
Working:
Using $1 + \tan^2 \theta = \sec^2 \theta$:
In QIV, $\sec \theta > 0$, so $\sec \theta = \frac{5}{3}$, giving $\cos \theta = \frac{3}{5}$.
Using $\tan \theta = \frac{\sin \theta}{\cos \theta}$:
Check: $\frac{16}{25} + \frac{9}{25} = 1$ ✓
Answer: $\sin \theta = \mathbf{-\frac{4}{5}}$, $\cos \theta = \mathbf{\frac{3}{5}}$
Simplify using identities
Simplify $\frac{\sin^2 \theta}{1 - \cos \theta}$ to a single trigonometric expression. Justify each step. (3 marks)
View comprehensive answer
Working:
Replace $\sin^2 \theta$ with $1 - \cos^2 \theta$:
Cancel (where $\cos \theta \neq 1$):
Answer: $\mathbf{1 + \cos \theta}$ (where $\cos \theta \neq 1$)
Key insight: recognising $1 - \cos^2 \theta$ as a difference of squares is the critical step.
Five timed questions on Pythagorean identities. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
Enter the arenaClimb platforms, hit checkpoints, and answer trig identity questions. Quick recall, lighter than the boss.
Mark lesson as complete
Tick when you've finished the practice and review.