An identity is true for every angle. Proving one means showing the two sides are the same expression written differently.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The one rule
Work on one side only. Moving terms across is what you do to an equation you are solving, and it assumes the very thing you are proving.
Three families
Reciprocal: csc = 1/sin. Quotient: tan = sin/cos. Pythagorean: sin²θ + cos²θ = 1.
A reliable strategy
Start on the messier side. Rewrite everything as sine and cosine, then simplify.
Two useful rearrangements
Dividing sin²θ + cos²θ = 1 by cos²θ gives tan²θ + 1 = sec²θ. Dividing by sin²θ gives 1 + cot²θ = csc²θ.
Work one side into the other
Start with the more complicated side and transform it until it matches the other. Working on both sides simultaneously risks assuming what you are trying to prove, which is not a valid argument.
The standard moves
Write everything in sines and cosines; find a common denominator; factor; substitute a Pythagorean identity; multiply by a conjugate. Almost every identity proof is a short sequence of these.
An identity is not an equation to solve
An identity holds for every permissible value, so there is nothing to solve for. Proving one means demonstrating the equality in general, which is a different task from finding which values satisfy an equation.
Note where it fails to apply
An identity involving tan θ says nothing at θ = π/2, where tan is undefined. A complete statement excludes those values. Ignoring the excluded points is a small dishonesty that matters in calculus.
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 Yusuf Prove tan θ · cos θ = sin θ
Yusuf must show the left side equals the right for every angle.
- Step 1
He starts on the left, which is the messier side.
Step 4: Your Turn
Practice makes it stick.
The Substitution
Problem 1 of 2
If sin²θ = 0.36, what is cos²θ?
The First Move
Problem 2 of 2
To prove sec θ · cot θ = csc θ, what is the best opening move?
Simplify and Prove
1 of 8
sin²θ = 0.49. What is cos²θ?
2 of 8
tan²θ + 1 = 9. What is sec²θ?
3 of 8
cot²θ = 15. What is csc²θ?
4 of 8
Match each expression with its simplest form.
Tap a card on the left to start.
5 of 8
cos θ · sec θ equals what number?
6 of 8
sec²θ − tan²θ equals what number?
7 of 8
Put the proof of tan θ · cos θ = sin θ in order.
- 1Replace tan θ with sin θ ÷ cos θ.
- 2Cancel the cos θ terms.
- 3The left side is now sin θ, matching the right.
- 4Start with the left side, tan θ · cos θ.
8 of 8
cos θ = 0.8 and θ is acute. What is sin θ?
Step 5: Quick Check
Show what you know.
Question 1 of 2
What is sin²θ + cos²θ for any angle?
Question 2 of 2
Why may you not move terms across an identity you are proving?
What You Learned
- An identity holds for every angle.
- Work one side only, starting with the messier one.
- Rewriting everything as sine and cosine almost always opens the way.