Trigonometry Identities & Equations Quiz, Explanations and Step-by-Step Review
Use the question set below to practice trigonometry identities and equations with high-impact skills: unit circle values and exact angles, Pythagorean identities \(\bigl(\sin^2\theta+\cos^2\theta=1,\;1+\tan^2\theta=\sec^2\theta,\;1+\cot^2\theta=\csc^2\theta\bigr)\), reciprocal identities and quotient identities, even-odd identities (negative angles), periodicity and phase shift identities (like \(\theta+2\pi\) and \(\theta+\pi\)), cofunction identities, sum and difference formulas for \(\sin\), \(\cos\), and \(\tan\), double-angle and half-angle identities, sum-to-product and product-to-sum transformations, and solving trigonometric equations on standard intervals such as \([0,2\pi)\). If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
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How this trigonometry practice works
1. Take the practice set: answer the trigonometry identities and equations questions below.
2. Open the lesson (optional): review core identities, transformations, and equation-solving strategies with worked examples.
3. Retry: return to the question set and apply the correct identity or solving step immediately.
What you will learn in the trigonometry identities & equations lesson
Identity foundations
Unit circle interpretation of \(\sin\theta\) and \(\cos\theta\)
Reciprocal and quotient identities: \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\), \(\sec\theta=\dfrac{1}{\cos\theta}\), etc.
Even-odd and periodicity rules for negative angles and shifts like \(\theta+2\pi\) and \(\theta+\pi\)
Pythagorean & shift identities
Pythagorean identities and how to rewrite everything in \(\sin\) and \(\cos\)
Purpose: Build a clear understanding of trigonometric identities and equations so you can simplify expressions, verify identities, and solve trigonometric equations efficiently. You’ll practice unit circle values, Pythagorean/reciprocal/quotient identities, symmetry and periodicity, angle sum and difference formulas, double-angle and half-angle identities, and sum-to-product / product-to-sum transformations.
Success criteria
Use radians and unit circle ideas to interpret \(\sin\theta\) and \(\cos\theta\).
Apply reciprocal and quotient identities: \(\sec\theta=\dfrac{1}{\cos\theta}\), \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\), etc.
Use Pythagorean identities to rewrite and simplify: \(\sin^2\theta+\cos^2\theta=1\), \(1+\tan^2\theta=\sec^2\theta\), \(1+\cot^2\theta=\csc^2\theta\).
Use even–odd identities and periodicity: \(\sin(-\theta)=-\sin\theta\), \(\cos(-\theta)=\cos\theta\), \(\tan(\theta+\pi)=\tan\theta\), \(\cos(\theta+2\pi)=\cos\theta\).
Expand or combine angles using sum and difference formulas for \(\sin\), \(\cos\), and \(\tan\).
Choose the best double-angle or half-angle form to simplify quickly.
Convert between sums and products using sum-to-product and product-to-sum identities.
Solve trigonometric equations on a specified interval (like \([0,2\pi)\)) and check solutions.
Key vocabulary
Identity: an equation true for all \(\theta\) in its domain (example: \(\sin^2\theta+\cos^2\theta=1\)).
Equation: an equality that is true only for certain angles (example: \(\sin x=\tfrac12\)).
Unit circle: the circle of radius \(1\); \((\cos\theta,\sin\theta)\) is a point on it.
Periodicity: repeating values, such as \(\sin(\theta+2\pi)=\sin\theta\).
Reference angle: the acute angle to the \(x\)-axis used to find exact trig values.
Solution set: all angles \(x\) that satisfy an equation within a given interval.
Quick pre-check
Pre-check 1: What is \(\sin(-\theta)\) equal to?
Hint: \(\sin\) is an odd function.
Pre-check 2: What is \(\cos(\theta+2\pi)\) equal to?
Hint: \(\cos\) has period \(2\pi\): shifting by \(2\pi\) returns the same value.
Identity Foundations
Core trigonometric identities: symmetry, periodicity, and Pythagorean structure
Learning goal: Recognize the core identity “toolbox” so you can simplify quickly and rewrite everything in a consistent form.
Key idea
Many identities come from the unit circle, where \((\cos\theta,\sin\theta)\) lies on a circle of radius \(1\). That immediately gives the Pythagorean identity: \[ \sin^2\theta+\cos^2\theta=1. \] From it, you can derive two useful variants by dividing by \(\cos^2\theta\) or \(\sin^2\theta\): \[ 1+\tan^2\theta=\sec^2\theta,\qquad 1+\cot^2\theta=\csc^2\theta. \] Also remember: reciprocal identities \(\sec\theta=\dfrac{1}{\cos\theta}\), \(\csc\theta=\dfrac{1}{\sin\theta}\), and quotient identities \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\), \(\cot\theta=\dfrac{\cos\theta}{\sin\theta}\).
Use the difference of squares: \[ (\sec x+\tan x)(\sec x-\tan x)=\sec^2x-\tan^2x. \] Now use the Pythagorean identity \(1+\tan^2x=\sec^2x\), which rearranges to \(\sec^2x-\tan^2x=1\). So the product simplifies to \(1\).
Try it
Try it 1: What is \(\tan(-\theta)\) equal to?
Hint: \(\tan\) is an odd function: \(\tan(-\theta)=-\tan\theta\).
Try it 2: What is \(\tan(\pi+\theta)\) equal to?
Hint: \(\tan\) has period \(\pi\), so shifting by \(\pi\) keeps the value the same.
Symmetry + periodicity are fast simplification tools for negative angles and shifts.
Sum & Difference
Angle sum and difference formulas
Learning goal: Expand \(\sin(A\pm B)\), \(\cos(A\pm B)\), and \(\tan(A\pm B)\) correctly and use these formulas to simplify expressions.
Key idea
The angle addition and subtraction identities let you rewrite trig functions of compound angles: \[ \sin(A\pm B)=\sin A\cos B \pm \cos A\sin B, \] \[ \cos(A\pm B)=\cos A\cos B \mp \sin A\sin B, \] \[ \tan(A\pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}. \] A common check: in the cosine identity, the sign flips (minus for \(+\), plus for \(-\)).
Worked example
Example: Write \(\cos(\alpha-\beta)\) in terms of \(\sin\alpha,\cos\alpha,\sin\beta,\cos\beta\).
Use the cosine difference formula: \[ \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta. \] This identity is especially useful for simplifying expressions and for proofs.
Try it
Try it 1: What is the formula for \(\tan(A-B)\)?
Hint: For \(\tan(A-B)\), the denominator uses \(1+\tan A\tan B\).
Try it 2: What is \(\cos(\alpha-\beta)\) equal to?
Hint: \(\cos(A-B)\) uses a plus between the \(\sin A\sin B\) term.
Summary
Sum/difference formulas expand compound angles into products of simpler trig functions.
These formulas are key for exact values, simplification, and identity proofs.
Double & Half Angle
Double-angle, half-angle, and smart rewrites
Learning goal: Use double-angle and half-angle identities to rewrite expressions in the most helpful form.
Key idea
Double-angle identities come from the sum formulas by setting \(A=B\): \[ \sin(2x)=2\sin x\cos x, \] \[ \cos(2x)=\cos^2x-\sin^2x=1-2\sin^2x=2\cos^2x-1, \] \[ \tan(2x)=\frac{2\tan x}{1-\tan^2x}. \] Choosing the best form of \(\cos(2x)\) depends on what you want to eliminate: use \(1-2\sin^2x\) if you want only \(\sin x\), or \(2\cos^2x-1\) if you want only \(\cos x\).
\(\tan\left(\dfrac{x}{2}\right)=\dfrac{\sin x}{1+\cos x}=\dfrac{1-\cos x}{\sin x}\) (when the denominators are nonzero)
Worked example
Example: Rewrite \(\cos(2x)\) using only \(\sin x\).
Start from \(\cos(2x)=\cos^2x-\sin^2x\). Using \(\cos^2x=1-\sin^2x\), we get: \[ \cos(2x)=(1-\sin^2x)-\sin^2x=1-2\sin^2x. \] So \(\cos(2x)\) can be written as \(1-2\sin^2x\).
Try it
Try it 1: What is the double-angle identity for \(\cos(2x)\)?
Hint: \(\cos(2x)\) has multiple equivalent forms; one classic form is \(\cos^2x-\sin^2x\).
Try it 2: What is \(\sin(2\theta)\) equal to in terms of \(\tan(\theta)\)?
Hint: \(\sin(2\theta)=2\sin\theta\cos\theta\), then convert using \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) and \(\sec^2\theta=1+\tan^2\theta\).
Summary
Double-angle identities are “high leverage” for simplification and equation solving.
Half-angle formulas help convert squares into linear trig expressions.
Sum/Product Tools
Sum-to-product and product-to-sum identities
Learning goal: Convert sums into products (for factoring) and products into sums (common in calculus and signal work).
Key idea
These transformations are especially useful when you want to factor or combine trig terms. Two key sum-to-product formulas are: \[ \sin A+\sin B=2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right), \] \[ \sin A-\sin B=2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right). \] (There are similar formulas for \(\cos A\pm \cos B\), and product-to-sum formulas for \(\sin A\sin B\), \(\cos A\cos B\), etc.)
Worked example
Example: Factor \(\sin x+\sin(3x)\) using a sum-to-product identity.
Use \(\sin A+\sin B=2\sin\left(\dfrac{A+B}{2}\right)\cos\left(\dfrac{A-B}{2}\right)\) with \(A=x\), \(B=3x\): \[ \sin x+\sin(3x)=2\sin\left(\frac{x+3x}{2}\right)\cos\left(\frac{x-3x}{2}\right)=2\sin(2x)\cos(-x). \] Since \(\cos(-x)=\cos x\), this becomes \(2\sin(2x)\cos x\).
Try it
Try it 1: Which identity transforms \(\sin A+\sin B\) into a product?
Hint: “Sum-to-product” turns a sum into a product with half-sum and half-difference angles.
Try it 2: What is the identity for \(\sin x-\sin y\)?
Hint: For \(\sin x-\sin y\), use \(2\cos\left(\dfrac{x+y}{2}\right)\sin\left(\dfrac{x-y}{2}\right)\).
Summary
Sum-to-product helps with factoring and equation solving.
Product-to-sum often appears in integration and simplifying products.
Solving Equations
Solving trigonometric equations on \([0,2\pi)\)
Learning goal: Solve trig equations systematically using identities, factoring, and the unit circle — then report solutions in the requested interval.
Core strategy
1) Simplify first: use identities to rewrite everything in a consistent form (often in \(\sin\) and \(\cos\)).
2) Isolate the trig function: aim for \(\sin x=c\), \(\cos x=c\), or \(\tan x=c\).
3) Use the unit circle: find all angles in the interval that match the value.
4) Check domain issues: avoid invalid steps like dividing by something that could be \(0\).
Worked example
Example: Solve for \(x\) in \([0,2\pi)\): \(\sin(2x)=\cos(2x)\).
Bring everything to one side: \[ \sin(2x)-\cos(2x)=0. \] If \(\cos(2x)≠ 0\), divide by \(\cos(2x)\) to get: \[ \tan(2x)=1. \] So \[ 2x=\frac{\pi}{4}+k\pi \quad \Rightarrow \quad x=\frac{\pi}{8}+k\frac{\pi}{2}. \] Now list solutions in \([0,2\pi)\): \[ x\in\left\{\frac{\pi}{8},\frac{5\pi}{8},\frac{9\pi}{8},\frac{13\pi}{8}\right\}. \] (You can also solve without dividing by rewriting as \(\sin(2x)=\cos(2x)=\frac{\sqrt2}{2}\) times a common factor, but \(\tan(2x)=1\) is the fastest route.)
Try it
Try it 1: Solve for \(x\) in \([0,2\pi)\): \(\sin x=\tfrac{\sqrt2}{2}\).
Hint: \(\sin x=\tfrac{\sqrt2}{2}\) happens at reference angle \(\pi/4\) in Quadrants I and II.
Try it 2: Solve for \(x\) in \([0,2\pi)\): \(2\sin x-1=0\).
Hint: Solve \(2\sin x-1=0\Rightarrow \sin x=\tfrac12\), then use the unit circle in \([0,2\pi)\).
Summary
Always respect the interval and list every solution inside it.
Use identities to simplify first; then use unit circle patterns to finish quickly.
Verify & Simplify
Verifying identities and simplifying expressions
Learning goal: Simplify confidently and verify identities without introducing invalid steps.
Best-practice checklist
Work from one side: start with the more complicated side and rewrite until it matches the other side.
Convert to \(\sin\) and \(\cos\) when stuck: many identities become straightforward after rewriting \(\tan\), \(\sec\), etc.
Use Pythagorean identities strategically: replace \(\sin^2\) with \(1-\cos^2\) or \(\sec^2\) with \(1+\tan^2\).
Avoid dividing by expressions that could be zero: if you divide by \(\sin x\), you are implicitly assuming \(\sin x≠ 0\).
Use double-angle identities: \[ 1-\cos(2x)=1-(1-2\sin^2x)=2\sin^2x, \] and \[ \sin(2x)=2\sin x\cos x. \] So \[ \frac{1-\cos(2x)}{\sin(2x)}=\frac{2\sin^2x}{2\sin x\cos x}=\frac{\sin x}{\cos x}=\tan x, \] as long as \(\sin(2x)≠ 0\) (the original expression’s domain restriction).
Try it
Try it 1: Simplify \((\sec x+\tan x)(\sec x-\tan x)\).
Hint: It’s a difference of squares: \(\sec^2x-\tan^2x\), then use \(\sec^2x-\tan^2x=1\).
Try it 2: What is \(\tan^2(\theta)\) in terms of \(\sec^2(\theta)\)?
Hint: Rearrange \(1+\tan^2\theta=\sec^2\theta\).
Summary
Identity work is about clean rewrites — not “guessing the answer.”
Pythagorean relationships are the backbone of many simplifications.
Applications & Big Picture
Why trig identities and equations matter
Learning goal: Connect the identity toolkit to real problem-solving — then finish with a final check.
Where these skills show up
Calculus: simplifying expressions before differentiating/integrating; using product-to-sum in integrals.
Geometry: triangle relationships, rotations, and periodic motion.
Signals: combining sinusoidal waves using sum-to-product identities.
Worked example: a fast shift simplification
Example: Simplify \(\cos(x+\pi)\) and explain what it means.
A shift by \(\pi\) flips the sign of cosine: \[ \cos(x+\pi)=-\cos x. \] Interpretation: adding \(\pi\) radians is a half-turn on the unit circle, sending \((\cos x,\sin x)\) to \((-\cos x,-\sin x)\).
Try it
Try it 1: What is \(\csc\!\left(\tfrac{\pi}{2}+\theta\right)\) equal to?
Hint: \(\sin\!\left(\tfrac{\pi}{2}+\theta\right)=\cos\theta\), so \(\csc\!\left(\tfrac{\pi}{2}+\theta\right)=\dfrac{1}{\cos\theta}=\sec\theta\).
Try it 2: What is the identity for \(\cos(x+\pi)\)?
Hint: Adding \(\pi\) radians is a half-turn on the unit circle, which flips the \(x\)-coordinate (cosine).
Equations: simplify, isolate, use the unit circle, respect the interval, and check restrictions.
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the identity or equation skill you need.
