Core trigonometric identities: symmetry, periodicity, and Pythagorean structure

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 \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\), \(\cot\theta=\dfrac{\cos\theta}{\sin\theta}\).

Symmetry & shift shortcuts

• Even–odd: \(\cos(-\theta)=\cos\theta\), \(\sin(-\theta)=-\sin\theta\), \(\tan(-\theta)=-\tan\theta\).
• Periodicity: \(\sin(\theta+2\pi)=\sin\theta\), \(\cos(\theta+2\pi)=\cos\theta\), \(\tan(\theta+\pi)=\tan\theta\).
• Shift by \(\pi\): \(\sin(\theta+\pi)=-\sin\theta\), \(\cos(\theta+\pi)=-\cos\theta\).

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Summary

• Pythagorean core: \(\sin^2\theta+\cos^2\theta=1\), \(1+\tan^2\theta=\sec^2\theta\), \(1+\cot^2\theta=\csc^2\theta\).
• Symmetry + periodicity are fast simplification tools for negative angles and shifts.

Angle sum and difference formulas

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 \(-\)).

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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-angle, half-angle, and smart rewrites

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 \(\cos x\).

Half-angle (power reduction) snapshots

• \(\sin^2\left(\dfrac{x}{2}\right)=\dfrac{1-\cos x}{2}\)
• \(\cos^2\left(\dfrac{x}{2}\right)=\dfrac{1+\cos x}{2}\)
• \(\tan\left(\dfrac{x}{2}\right)=\dfrac{\sin x}{1+\cos x}=\dfrac{1-\cos x}{\sin x}\) (when the denominators are nonzero)

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Summary

• Double-angle identities are “high leverage” for simplification and equation solving.
• Half-angle formulas help convert squares into linear trig expressions.

Sum-to-product and product-to-sum identities

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.)

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Summary

• Sum-to-product helps with factoring and equation solving.
• Product-to-sum often appears in integration and simplifying products.

Solving trigonometric equations on \([0,2\pi)\)

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 \(0\).

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Summary

• Always respect the interval and list every solution inside it.
• Use identities to simplify first; then use unit circle patterns to finish quickly.

Verifying identities and simplifying expressions

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.

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Summary

• Identity work is about clean rewrites — not “guessing the answer.”
• Pythagorean relationships are the backbone of many simplifications.

Why trig identities and equations matter

Where these skills show up

• Calculus: simplifying expressions before differentiating/integrating; using product-to-sum in integrals.
• Physics & engineering: resolving forces, oscillations, wave models, and phase shifts.
• Geometry: triangle relationships, rotations, and periodic motion.
• Signals: combining sinusoidal waves using sum-to-product identities.

Worked example: a fast shift simplification

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Final recap

• Foundation: reciprocal/quotient identities + Pythagorean identities (\(\sin^2+\cos^2=1\), \(1+\tan^2=\sec^2\)).
• Compound angles: sum/difference formulas for \(\sin\), \(\cos\), \(\tan\).
• Transforms: double-angle/half-angle + sum-to-product/product-to-sum.
• Equations: simplify, isolate, use the unit circle, respect the interval, and check restrictions.