Precalc
Section 7.1

Trig Identities

These identities are the tools you'll use in Sections 7.2, 7.3, and 7.5. You can't simplify equations or verify identities without them. Expect at least 2 identity problems on Exam 3 — one simplification and one 'verify' proof.

⚡ Quick Summary

A trig identity is an equation that's always true, no matter what angle you plug in — like a universal shortcut. The most important one is sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 (the Pythagorean identity). Think of it like this: on the unit circle, every point satisfies x2+y2=1x^2 + y^2 = 1, and since x=cos⁡θx = \cos\theta and y=sin⁡θy = \sin\theta, the identity writes itself. You'll rearrange it constantly — sin⁡2=1−cos⁡2\sin^2 = 1 - \cos^2 and cos⁡2=1−sin⁡2\cos^2 = 1 - \sin^2 are just as useful.

Pythagorean Identity
sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1

The most important identity. Rearranges to give you sin² or cos² alone.

Pythagorean Identity (tan/sec)
1+tan⁡2(θ)=sec⁡2(θ)1 + \tan^2(\theta) = \sec^2(\theta)

Divide the main Pythagorean identity by cos²(θ) to get this one.

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

A trig identity is an equation that's always true, no matter what angle you plug in — like a universal shortcut. The most important one is sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 (the Pythagorean identity). Think of it like this: on the unit circle, every point satisfies x2+y2=1x^2 + y^2 = 1, and since x=cos⁡θx = \cos\theta and y=sin⁡θy = \sin\theta, the identity writes itself. You'll rearrange it constantly — sin⁡2=1−cos⁡2\sin^2 = 1 - \cos^2 and cos⁡2=1−sin⁡2\cos^2 = 1 - \sin^2 are just as useful.

The main skill here is rewriting trig expressions into simpler forms. If you see tangent, secant, cosecant, or cotangent in a complicated expression, the first thing to try is converting everything to sine and cosine. Things usually start canceling. If that doesn't work, look for Pythagorean identity patterns or try factoring.

When the exam says "verify the identity," it means prove that both sides of an equation are equal — but with a special rule: you can only work on one side at a time. Pick the messier-looking side and simplify it step by step until it matches the other side. Think of the equals sign as a wall you can't reach across. Your three go-to moves are: (1) rewrite in sine/cosine, (2) factor, (3) multiply by a conjugate like 1+sin⁡θ1 + \sin\theta.

This section is all about practice — the more identities you simplify, the faster you'll spot patterns. Don't worry if it feels slow at first. Everyone struggles with identities initially, and then it clicks.

Key Formulas

Pythagorean Identity
sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1

The most important identity. Rearranges to give you sin² or cos² alone.

Pythagorean Identity (tan/sec)
1+tan⁡2(θ)=sec⁡2(θ)1 + \tan^2(\theta) = \sec^2(\theta)

Divide the main Pythagorean identity by cos²(θ) to get this one.

Pythagorean Identity (cot/csc)
1+cot⁡2(θ)=csc⁡2(θ)1 + \cot^2(\theta) = \csc^2(\theta)

Divide the main Pythagorean identity by sin²(θ) to get this one.

Quotient Identities
tan⁡(θ)=sin⁡(θ)cos⁡(θ),cot⁡(θ)=cos⁡(θ)sin⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, \quad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
Reciprocal Identities
csc⁡(θ)=1sin⁡(θ),sec⁡(θ)=1cos⁡(θ),cot⁡(θ)=1tan⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}, \quad \sec(\theta) = \frac{1}{\cos(\theta)}, \quad \cot(\theta) = \frac{1}{\tan(\theta)}

Key Takeaways

  • ✓The Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 is the most-used identity — know its rearranged forms (sin⁡2=1−cos⁡2\sin^2 = 1 - \cos^2 and cos⁡2=1−sin⁡2\cos^2 = 1 - \sin^2)
  • ✓To simplify: rewrite everything in terms of sin⁡\sin and cos⁡\cos, then look for cancellations or Pythagorean patterns
  • ✓To verify an identity: work on one side only — never move terms across the equals sign
  • ✓Common strategies: factor, multiply by a conjugate (1+sin⁡θ1 + \sin\theta), or convert to sin/cos

⚠️ Common Mistakes

  • ✗Working on both sides of an identity at once — when verifying, only transform one side; treat the equals sign as a wall you can't cross
  • ✗Moving terms across the equals sign — this assumes the identity is true, which is exactly what you're trying to prove
  • ✗Forgetting the Pythagorean identity rearrangements: sin⁡2θ=1−cos⁡2θ\sin^2\theta = 1 - \cos^2\theta and cos⁡2θ=1−sin⁡2θ\cos^2\theta = 1 - \sin^2\theta are used just as often as the original
  • ✗Giving up too early — if rewriting in sin⁡\sin/cos⁡\cos doesn't simplify immediately, try factoring or multiplying by a conjugate like 1+sin⁡θ1 + \sin\theta
Next: Section 7.2Builds on: 6.3
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6.3 Inverse Trigonometric Functions
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7.2 Sum & Difference Identities