Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value in their domain. They are essential tools for simplifying expressions, solving equations, and evaluating non-standard angles.
This guide covers reciprocal, quotient, Pythagorean, co-function, even-odd, sum & difference, double angle, and half angle identities — with simplification strategies, worked examples, and a 10-question practice quiz.
1Introduction
While the unit circle and basic trig ratios let you evaluate specific angles, trigonometric identities let you transform and simplify expressions — replacing one form with an equivalent form. This is the algebraic backbone of trigonometry.
Identities are used constantly in calculus (integration techniques), physics (wave equations, harmonic motion), and engineering (signal processing, AC circuits). Mastering them now gives you a massive advantage in later courses.
In calculus, evaluating the integral of sin²x requires the power-reducing identity sin²x = (1 − cos 2x)/2. Without identities, many integrals and derivatives would be impossible to compute in closed form.
Physics & Engineering
Wave interference, AC circuit analysis, and harmonic motion all rely on sum/difference and double angle formulas.
Computer Graphics
Rotation matrices use cos and sin identities to rotate points and objects in 2D and 3D space.
2Key Definitions
Identity
An equation true for all values of the variable in its domain. Example: sin²x + cos²x = 1.
Conditional Equation
An equation true only for specific values. Example: sin x = 1/2 (true at x = 30°, 150°, etc.).
sin θ
Opposite / Hypotenuse (or y-coordinate on unit circle).
cos θ
Adjacent / Hypotenuse (or x-coordinate on unit circle).
tan θ
Opposite / Adjacent = sin θ / cos θ.
csc θ
1 / sin θ (reciprocal of sine).
sec θ
1 / cos θ (reciprocal of cosine).
cot θ
1 / tan θ = cos θ / sin θ (reciprocal of tangent).
3Reciprocal & Quotient Identities
These identities define the relationships between the six trigonometric functions. They are the most basic identities and are used constantly in simplifications.
Reciprocal Identities
csc θ
= 1 / sin θ
sec θ
= 1 / cos θ
cot θ
= 1 / tan θ
Quotient Identities
tan θ
= sin θ / cos θ
cot θ
= cos θ / sin θ
Strategy: Convert to Sine and Cosine
When simplifying a complex expression, your first move should almost always be to rewrite everything in terms of sin and cos using these identities. This makes Pythagorean identity applications much easier to spot.
4Pythagorean Identities
These three identities are derived from the Pythagorean theorem applied to the unit circle. The first is the most fundamental identity in trigonometry; the other two are obtained by dividing by cos²θ or sin²θ.
The Three Pythagorean Identities
Rearranged Forms
Each Pythagorean identity can be rearranged. Recognizing these alternative forms is critical for simplification.
From Identity 1
sin²θ = 1 − cos²θ
cos²θ = 1 − sin²θ
From Identity 2
tan²θ = sec²θ − 1
sec²θ − tan²θ = 1
From Identity 3
cot²θ = csc²θ − 1
csc²θ − cot²θ = 1
5Co-function & Even-Odd Identities
Co-function Identities
Co-functions relate a function to its complement. The prefix "co-" in cosine, cotangent, and cosecant means "complement."
Even-Odd Identities
Even functions satisfy f(−x) = f(x). Odd functions satisfy f(−x) = −f(x). Cosine and secant are the only even trig functions.
Even (unchanged)
cos(−θ) = cos θ
sec(−θ) = sec θ
Odd (sign flips)
sin(−θ) = −sin θ
tan(−θ) = −tan θ
csc(−θ) = −csc θ
cot(−θ) = −cot θ
Memory Trick: "All Students Take Calculus"
This mnemonic tells you which functions are positive in each quadrant: All (QI), Sine (QII), Tangent (QIII), Cosine (QIV). Combined with even-odd identities, you can evaluate any trig function of a negative angle.
6Sum & Difference Formulas
These formulas let you find exact values of non-standard angles by breaking them into sums or differences of known angles (30°, 45°, 60°, 90°).
Sine Sum & Difference
sin(A + B) = sin A cos B + cos A sin B
sin(A − B) = sin A cos B − cos A sin B
Cosine Sum & Difference
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
Tangent Sum & Difference
tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
tan(A − B) = (tan A − tan B) / (1 + tan A tan B)
Watch the Signs!
In the cosine formulas, the signs are opposite to what you might expect: cos(A + B) uses a minus sign between terms, and cos(A − B) uses a plus. This is the opposite of sine, where the signs match.
7Double & Half Angle Formulas
Double angle formulas are derived by setting A = B in the sum formulas. Half angle formulas are derived from the double angle formulas by solving for the half-angle term.
Double Angle Formulas
sin(2θ)
= 2 sin θ cos θ
cos(2θ) — three equivalent forms:
= cos²θ − sin²θ
= 2cos²θ − 1
= 1 − 2sin²θ
tan(2θ)
= 2tan θ / (1 − tan²θ)
Half Angle Formulas
sin(θ/2)
= ±√((1 − cos θ) / 2)
cos(θ/2)
= ±√((1 + cos θ) / 2)
tan(θ/2)
= sin θ / (1 + cos θ) = (1 − cos θ) / sin θ
Power-Reducing Formulas
Rearranging the double angle formulas for cos(2θ) gives power-reducing identities — essential for integration in calculus.
8Simplification Strategies
Follow this systematic approach when simplifying or verifying trigonometric expressions.
The 6-Step Simplification Process
- Work on one side only (when verifying an identity). Never cross the equals sign.
- Convert to sine and cosine. Replace tan, cot, sec, csc using reciprocal/quotient identities.
- Combine fractions by finding a common denominator.
- Factor when possible — look for difference of squares (a² − b²) and common factors.
- Apply Pythagorean identities to replace sin²x + cos²x with 1 (or rearranged forms).
- Simplify by canceling common factors.
Look for Conjugates
Multiplying by (1 + sin x)/(1 + sin x) or similar conjugates often unlocks difference-of-squares patterns like 1 − sin²x = cos²x.
Try Substitution
If you see sec²x − 1, recognize it as tan²x. Train yourself to spot these patterns in both directions.
9Worked Examples
Example 1: Simplify tan²x − sec²x + 1
Answer: 0
Example 2: Find the exact value of cos(75°)
Answer: (√6 − √2)/4
Example 3: Verify the identity (1 − cos²x) / sin x = sin x
Example 4: If sin θ = 5/13 and θ is in QII, find sin(2θ)
Answer: −120/169
10Memory Aids
"Sohcahtoa + Cho Sha Cao"
SOH-CAH-TOA gives primary ratios. For reciprocals: Csc = Hyp/Opp, Sec = Hyp/Adj, Cot = Adj/Opp.
Cosine Sum: Signs Are Opposite
cos(+) uses a minus. cos(−) uses a plus. The signs in the formula are always opposite to the operation between A and B.
sin(2θ) = "2 Sinners Cossing"
sin(2θ) = 2 sinθ cosθ. Remember: the double angle sine formula always has a 2 in front.
Even = Cosine (both have "e")
Cosine and secant are even functions. Everything else (sin, tan, csc, cot) is odd.
11Common Mistakes
sin(A + B) ≠ sin A + sin B
Trig functions do NOT distribute over addition. You must use the sum formula: sin(A + B) = sin A cos B + cos A sin B.
Confusing sin²x with sin(x²)
sin²x means (sin x)² — square the output. sin(x²) means take the sine of x² — square the input. These are very different values.
Wrong Sign in Cosine Sum Formula
cos(A + B) = cos A cos B − sin A sin B. Students often write a plus sign here because the sum formula for sine uses plus.
Forgetting the ± in Half Angle Formulas
The half angle formulas have ± in front of the square root. You must determine the correct sign based on the quadrant where θ/2 lies.
Working Both Sides of an Identity
When verifying an identity, work on ONE side only. If you manipulate both sides simultaneously, you are assuming the identity is true — which is what you are trying to prove.
Quick Revision
- Reciprocal: csc = 1/sin, sec = 1/cos, cot = 1/tan.
- Quotient: tan = sin/cos, cot = cos/sin.
- Pythagorean: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ.
- Co-function: sin(π/2 − θ) = cos θ and vice versa.
- Even: cos(−θ) = cos θ, sec(−θ) = sec θ. Odd: sin(−θ) = −sin θ, tan(−θ) = −tan θ.
- Sum: sin(A ± B) = sin A cos B ± cos A sin B. cos(A ± B) = cos A cos B ∓ sin A sin B.
- Double angle: sin(2θ) = 2sin θ cos θ. cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
- Half angle: sin(θ/2) = ±√((1 − cos θ)/2). cos(θ/2) = ±√((1 + cos θ)/2).
- Strategy: Convert to sin/cos → combine fractions → factor → apply Pythagorean → simplify.
FAQ
What is the difference between an identity and an equation?
An identity is true for ALL values in its domain (e.g., sin²x + cos²x = 1). An equation is true only for specific values (e.g., sin x = 1/2 is only true at x = π/6, 5π/6, etc.).
Do I need to memorize all these identities?
Focus on: the three Pythagorean identities, reciprocal/quotient identities, and the sum formulas for sine and cosine. You can derive the double-angle and half-angle formulas from these.
When should I use identities vs. the unit circle?
Use the unit circle to evaluate specific angles (sin 30°, cos π/4). Use identities to simplify expressions, verify equations, or find exact values of non-standard angles like 75° or 15°.
How do I prove/verify a trigonometric identity?
Work on ONE side only (usually the more complex side). Convert everything to sine and cosine, look for common factors, and use Pythagorean identities to simplify.
14Practice Quiz
Test your knowledge — select the correct answer for each question.
1.Which of the following is equivalent to 1/tan θ?
2.Simplify: cos²x + sin²x
3.If sin θ = 0.6, what is cos(π/2 − θ)?
4.Which identity is used to derive 1 + cot²θ = csc²θ from sin²θ + cos²θ = 1?
5.What is the value of sin(−θ) if sin θ = 0.5?
6.The expression sin θ / cos θ is equivalent to:
7.Which of these is NOT a correct form for cos(2θ)?
8.Find the exact value of sin(105°). (Hint: 105° = 60° + 45°)
9.If cos θ = 4/5 and θ is in Quadrant IV, what is sin(2θ)?
10.Simplify: (sec x − tan x)(sec x + tan x)
Study Tips
- Start by memorizing the Pythagorean identity and reciprocal identities — everything else builds on them.
- Practice deriving double angle from sum formulas to deepen understanding.
- When stuck on a verification, try converting everything to sine and cosine first.
- Create flashcards with the identity on one side and a worked example on the other.
- Work practice problems daily — pattern recognition improves with repetition.