Trig Identities Calculator — Verify, Prove & Simplify with Steps

Trigonometry Tool

Trig Identities Calculator

Q: What are the three Pythagorean trig identities?

The three Pythagorean identities are: (1) sin²θ + cos²θ = 1 — the fundamental identity derived from the unit circle; (2) tan²θ + 1 = sec²θ — obtained by dividing the first identity by cos²θ; (3) 1 + cot²θ = csc²θ — obtained by dividing the first identity by sin²θ.

Q: How do you verify a trig identity?

To verify a trig identity: (1) Work on ONE side only — never move terms across the equals sign. (2) Start with the more complex side. (3) Convert everything to sin and cos. (4) Look for Pythagorean substitutions (sin²+cos²=1, etc.). (5) Factor, simplify, and use algebraic techniques. (6) Stop when both sides match.

Q: What is the difference between a trig identity and a trig equation?

A trig identity is true for ALL values of the variable — e.g., sin²θ + cos²θ = 1 holds for every angle. A trig equation is true only for specific values — e.g., sin(x) = 0.5 is only true for x = 30° and x = 150° (plus multiples of 360°).

Q: How do you solve trig equations?

To solve a trig equation like sin(x) = 0.5: (1) Isolate the trig function. (2) Find the reference angle using the inverse function: arcsin(0.5) = 30°. (3) Determine which quadrants give the correct sign (sin positive → Q1 and Q2). (4) Find all solutions in [0°, 360°): 30° and 150°. (5) Write the general solution: x = 30° + 360°n or x = 150° + 360°n.

Q: What are the double angle formulas?

The double angle formulas are: sin(2θ) = 2sinθcosθ; cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ (three equivalent forms); tan(2θ) = 2tanθ/(1 − tan²θ). These are derived from the sum formulas by setting A = B = θ.

Q: What is the half angle formula for sin?

The half angle formula for sin is: sin(θ/2) = ±√((1 − cosθ)/2). The ± sign depends on which quadrant θ/2 falls in — positive if θ/2 is in Q1 or Q2, negative if in Q3 or Q4. This is derived by solving cos(2α) = 1 − 2sin²α for sinα, then replacing α with θ/2.

Verify and prove trigonometric identities numerically and algebraically with step-by-step working, solve trig equations using the reference-angle method, and search the complete reference of all standard trig identities — Pythagorean, double angle, half angle, sum-to-product, and product-to-sum formulas. The ultimate tool for verifying trig identities and proving trigonometric identities.

Trig Identities Calculator — Four Tools

This verifier uses numerical testing at 12 angles — for absolute certainty, use the algebraic proof steps in the Prove tab below.

Left-Hand Side (LHS):

Use sin(x), cos(x), tan(x), sec(x), csc(x), cot(x), sin^2(x), ^ for powers, x or θ as variable.

sin²+cos²=1

Power reducing

Double angle sin

Quotient tan

tan²+1=sec²

(sin+cos)²

cos(2θ) form

Error

Test Values — LHS and RHS evaluated at 6 angles:

θ (rad)	LHS value	RHS value	\|Diff\|	Match

Select an identity to prove step-by-step:

Select an identity above to see its step-by-step proof

Trig Equation — format: sin(x) = 0.5

Units:

sin(x)=0.5

cos(x)=-1

tan(x)=1

sin(x)=-√2/2

cos(x)=0.5

tan(x)=-1

sin(x)=√3/2

Error

ASTC — All Students Take Calculus (which functions are positive):

sin +

Students

All +

All

tan +

Take

cos +

Calculus

Pythagorean Identities (3)

LHS	RHS	Notes
sin²θ + cos²θ	1	Fundamental — from unit circle
tan²θ + 1	sec²θ	Divide sin²+cos²=1 by cos²θ
1 + cot²θ	csc²θ	Divide sin²+cos²=1 by sin²θ

Reciprocal & Quotient Identities (8)

LHS	RHS	Type
sinθ	1/cscθ	Reciprocal
cosθ	1/secθ	Reciprocal
tanθ	1/cotθ	Reciprocal
cscθ	1/sinθ	Reciprocal
secθ	1/cosθ	Reciprocal
cotθ	cosθ/sinθ	Reciprocal
tanθ	sinθ/cosθ	Quotient
cotθ	cosθ/sinθ	Quotient

Sum & Difference Formulas (6)

LHS	RHS
sin(A+B)	sinA cosB + cosA sinB
sin(A−B)	sinA cosB − cosA sinB
cos(A+B)	cosA cosB − sinA sinB
cos(A−B)	cosA cosB + sinA sinB
tan(A+B)	(tanA + tanB)/(1 − tanA tanB)
tan(A−B)	(tanA − tanB)/(1 + tanA tanB)

Double Angle Formulas (4)

LHS	RHS	Form
sin(2θ)	2sinθ cosθ	Standard
cos(2θ)	cos²θ − sin²θ	Form 1
cos(2θ)	2cos²θ − 1	Form 2
cos(2θ)	1 − 2sin²θ	Form 3

Half Angle Formulas (3)

LHS	RHS	Note
sin(θ/2)	±√((1−cosθ)/2)	Sign by quadrant
cos(θ/2)	±√((1+cosθ)/2)	Sign by quadrant
tan(θ/2)	sinθ/(1+cosθ)	= (1−cosθ)/sinθ

Power Reducing Formulas (3)

LHS	RHS
sin²θ	(1 − cos2θ)/2
cos²θ	(1 + cos2θ)/2
tan²θ	(1 − cos2θ)/(1 + cos2θ)

Product-to-Sum Formulas (4)

LHS	RHS
sinA sinB	(1/2)[cos(A−B) − cos(A+B)]
cosA cosB	(1/2)[cos(A−B) + cos(A+B)]
sinA cosB	(1/2)[sin(A+B) + sin(A−B)]
cosA sinB	(1/2)[sin(A+B) − sin(A−B)]

Sum-to-Product Formulas (4)

LHS	RHS
sinA + sinB	2 sin((A+B)/2) cos((A−B)/2)
sinA − sinB	2 cos((A+B)/2) sin((A−B)/2)
cosA + cosB	2 cos((A+B)/2) cos((A−B)/2)
cosA − cosB	−2 sin((A+B)/2) sin((A−B)/2)

Co-function Identities (6)

LHS	RHS
sin(90°−θ)	cosθ
cos(90°−θ)	sinθ
tan(90°−θ)	cotθ
csc(90°−θ)	secθ
sec(90°−θ)	cscθ
cot(90°−θ)	tanθ

Even/Odd Identities (6)

LHS	RHS	Type
sin(−θ)	−sinθ	Odd
cos(−θ)	cosθ	Even
tan(−θ)	−tanθ	Odd
csc(−θ)	−cscθ	Odd
sec(−θ)	secθ	Even
cot(−θ)	−cotθ	Odd

What Are Trig Identities? — The Complete List

This trig identities calculator verifies and proves trigonometric identities numerically and algebraically — making it the fastest tool for verifying trig identities, checking your homework, and understanding why each identity is true. A trigonometric identity is an equation involving trig functions that holds true for every value of the variable (for which both sides are defined). Unlike a trig equation — which is true only for specific angles — a trig identity is a universal truth.

The trig identities fall into eleven families: Pythagorean identities (the most fundamental, derived from the unit circle), reciprocal and quotient identities (the definitions of tan, cot, sec, csc), sum and difference formulas, double angle formulas, half angle formulas, power reducing formulas, product-to-sum formulas, sum-to-product formulas, co-function identities, and even/odd identities. The complete searchable reference table is in the Reference tab above.

Why trig identities matter: They are used to simplify complex trig expressions, evaluate exact values (like sin 75°), solve trig equations, compute trig integrals (power-reducing formulas convert sin²θ into integrable form), and prove other results. Every calculus course depends on students mastering the Pythagorean trig identities.

Pythagorean Identities — The Three Forms

The three Pythagorean identities are the engine of almost every trig proof. When stuck on a trig identity problem, substituting using a Pythagorean identity is almost always the key move.

sin²θ + cos²θ = 1 The fundamental Pythagorean identity — derived from the unit circle equation x² + y² = 1

Derivation from the unit circle: On the unit circle, any point is (x, y) = (cosθ, sinθ). Since every point satisfies x² + y² = 1, we get cos²θ + sin²θ = 1 — the most fundamental of all trig identities.

Deriving the Second Pythagorean Identity: tan²θ + 1 = sec²θ

Start with sin²θ + cos²θ = 1 and divide every term by cos²θ:

sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ
tan²θ + 1 = sec²θ (since sin/cos = tan and 1/cos = sec)
tan²θ + 1 = sec²θ ✓

Deriving the Third Pythagorean Identity: 1 + cot²θ = csc²θ

Start: sin²θ + cos²θ = 1
Divide every term by sin²θ: 1 + cos²θ/sin²θ = 1/sin²θ
1 + cot²θ = csc²θ ✓

Identity	LHS	RHS	Derived by
Pythagorean 1	sin²θ + cos²θ	1	Unit circle
Pythagorean 2	tan²θ + 1	sec²θ	÷ cos²θ
Pythagorean 3	1 + cot²θ	csc²θ	÷ sin²θ

How to Verify a Trig Identity — Step-by-Step Method

Verifying trig identities requires a specific technique — you cannot treat them like algebraic equations. The rules for verifying trig identities are strict:

Work on ONE side only — never move terms across the equals sign
Start with the more complex side
Convert everything to sin and cos — replace tan, sec, csc, cot with their sin/cos equivalents
Look for Pythagorean substitutions — 1−sin²θ = cos²θ etc.
Factor, simplify, find common denominators
Multiply by conjugate for expressions like (1 ± sinθ)
Stop when both sides match

Example 1 — Verify: tanθ·cosθ = sinθ

Start with LHS: tanθ·cosθ
Replace tanθ with sinθ/cosθ: (sinθ/cosθ)·cosθ
Cancel cosθ: sinθ
LHS = sinθ = RHS ✓

Example 2 — Verify: sin²θ + cos²θ = 1 (Pythagorean)

This is the definition from the unit circle — no manipulation needed
Test at θ = 0.3: sin²(0.3) + cos²(0.3) = 0.08733 + 0.91267 = 1.00000 ✓
Identity confirmed by numerical and geometric proof ✓

Example 3 — Verify: (1 − cos²θ)/sin²θ = 1

Start with LHS: (1 − cos²θ)/sin²θ
Pythagorean substitution: 1 − cos²θ = sin²θ
sin²θ/sin²θ = 1
LHS = 1 = RHS ✓

Example 4 — Verify: secθ − sinθ·tanθ = cosθ

Convert to sin/cos: 1/cosθ − sinθ·(sinθ/cosθ)
Common denominator: (1 − sin²θ)/cosθ
Pythagorean: (cos²θ)/cosθ = cosθ
LHS = cosθ = RHS ✓

Example 5 — Verify: 1/(1−sinθ) − 1/(1+sinθ) = 2tanθ·secθ

Common denominator: [(1+sinθ)−(1−sinθ)] / [(1−sinθ)(1+sinθ)]
Numerator: 2sinθ; Denominator: 1−sin²θ = cos²θ
2sinθ/cos²θ = 2(sinθ/cosθ)·(1/cosθ) = 2tanθ·secθ
LHS = 2tanθ·secθ = RHS ✓

How to Prove Trig Identities — Common Strategies

Proving trigonometric identities means showing algebraically that LHS = RHS. This trigonometry proof solver approach uses eight standard strategies — mastering these lets you prove any identity:

Strategy 1 — Convert Everything to sin and cos

Replace tan, sec, csc, cot with their sin/cos definitions. Simplify the resulting expression. This works for nearly all identity proofs.

Strategy 2 — Pythagorean Substitution

Recognise sin²+cos²=1 patterns: if you see 1−sin²θ, replace with cos²θ. If you see 1+tan²θ, replace with sec²θ. These substitutions unlock most proofs.

Strategy 3 — Factor

Look for common factors: sin²θ + sinθcosθ = sinθ(sinθ + cosθ). Or difference of squares: sin²θ − cos²θ = (sinθ+cosθ)(sinθ−cosθ).

Strategy 4 — Common Denominator

When adding fractions: sinθ/cosθ + cosθ/sinθ → common denominator is sinθcosθ → (sin²θ+cos²θ)/(sinθcosθ) = 1/(sinθcosθ) = cscθ·secθ.

Strategy 5 — Multiply by Conjugate

For expressions like (1−cosθ), multiply and divide by (1+cosθ) to get (1−cos²θ) = sin²θ in the numerator.

Strategy 6 — Work from Both Sides

Simplify LHS to a middle expression; simplify RHS to the same middle expression. Show they meet. (Note: still document the proof going one direction only.)

Strategy 7 — Use Double Angle Formulas

Replace sin(2θ), cos(2θ), tan(2θ) with their expanded forms, or factor expanded trig expressions back into double angle form.

Strategy 8 — Power Reduction

Replace sin²θ with (1−cos2θ)/2 and cos²θ with (1+cos2θ)/2 to simplify squared trig functions in integrals and proofs.

Double Angle and Half Angle Formulas

The double angle formulas are derived from the sum formulas by setting A = B = θ. These trig identities are among the most useful in calculus.

sin(2θ) = 2 sinθ cosθ From sin(A+B) with A=B=θ: sinθcosθ + cosθsinθ = 2sinθcosθ

Three forms of cos(2θ): Choose the form that most simplifies your expression:

Form	Formula	Best used when
Form 1	cos²θ − sin²θ	Both sin and cos present
Form 2	2cos²θ − 1	Need to eliminate sin²θ
Form 3	1 − 2sin²θ	Need to eliminate cos²θ

The power-reducing formula sin²θ = (1−cos2θ)/2 is derived from Form 3: 1−2sin²θ = cos(2θ) → 2sin²θ = 1−cos(2θ) → sin²θ = (1−cos2θ)/2. This is essential for computing ∫sin²θ dθ in calculus.

Example — Simplify sin(2θ)/sinθ using double angle

Replace sin(2θ) = 2sinθcosθ: (2sinθcosθ)/sinθ
Cancel sinθ: 2cosθ
sin(2θ)/sinθ = 2cosθ ✓

How to Solve Trig Equations — Step-by-Step

The trig equation solver uses the reference-angle method. A trig equation like sin(x) = 0.5 is only true for specific angles — unlike a trig identity. Follow these five steps for any basic trig equation:

Isolate the trig function on one side
Find the reference angle using the inverse function (arcsin, arccos, arctan of the absolute value)
Determine the sign of the function value (positive or negative)
Use ASTC to find which quadrants have that sign
List solutions in [0°, 360°) and write the general solution

Example 1 — Solve: sin(x) = 0.5 for x ∈ [0°, 360°)

Reference angle = arcsin(0.5) = 30°
sin is positive → Q1 and Q2 (ASTC: S in Students)
Q1: x = 30° | Q2: x = 180° − 30° = 150°
Solutions: x = 30°, x = 150°
General: x = 30° + 360°n or x = 150° + 360°n

Example 2 — Solve: cos(x) = −1 for x ∈ [0°, 360°)

Reference angle = arccos(1) = 0°, but cos = −1 means we look for 180°
cos = −1 only at x = 180°
Solution: x = 180°
General: x = 180° + 360°n

Example 3 — Solve: tan(x) = 1 for x ∈ [0°, 360°)

Reference angle = arctan(1) = 45°
tan is positive → Q1 and Q3 (ASTC: A and T)
Q1: x = 45° | Q3: x = 180° + 45° = 225°
Solutions: x = 45°, x = 225°
General: x = 45° + 180°n (tan period is 180°, not 360°!)

Sum & Difference Formulas — Derivation and Examples

The six sum and difference formulas allow computation of exact trig values and are the source of most other trig identities.

Formula	Expansion
sin(A+B)	sinA cosB + cosA sinB
sin(A−B)	sinA cosB − cosA sinB
cos(A+B)	cosA cosB − sinA sinB
cos(A−B)	cosA cosB + sinA sinB
tan(A+B)	(tanA + tanB)/(1 − tanA tanB)
tan(A−B)	(tanA − tanB)/(1 + tanA tanB)

Example — Find sin(75°) exactly

Write 75° = 45° + 30°
sin(75°) = sin(45°+30°) = sin45°cos30° + cos45°sin30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
sin(75°) = (√6 + √2)/4 ≈ 0.9659

Example — Find cos(15°) exactly

Write 15° = 45° − 30°
cos(15°) = cos45°cos30° + sin45°sin30°
= (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4
cos(15°) = (√6 + √2)/4 ≈ 0.9659

Common Mistakes in Trig Identity Problems

Mistake 1 — Moving Terms Across the Equals Sign

❌ Wrong: sin²θ + cos²θ = 1 → "so sin²θ = 1 − cos²θ is the proof" (you rearranged, not proved)
✅ Correct: Work on one side only. Transform LHS into RHS using trig identities without touching the other side.

Mistake 2 — sin(A+B) ≠ sinA + sinB

❌ Wrong: sin(30°+60°) = sin30° + sin60° = 0.5 + 0.866 = 1.366 (impossible — sin ≤ 1)
✅ Correct: sin(90°) = 1. Use the sum formula: sin(A+B) = sinAcosB + cosAsinB

Mistake 3 — Using the Wrong cos(2θ) Form

When proving an identity, choose the form of cos(2θ) that matches what's already in the expression.
If the expression has sin²θ, use cos(2θ) = 1−2sin²θ. If it has cos²θ, use cos(2θ) = 2cos²θ−1.

Mistake 4 — Ignoring the ± in Half-Angle Formulas

❌ Wrong: sin(θ/2) = +√((1−cosθ)/2) always
✅ Correct: sin(θ/2) = ±√((1−cosθ)/2) — the sign depends on which quadrant θ/2 falls in.

Mistake 5 — Squaring Both Sides to Verify

❌ Wrong: Squaring both sides of LHS = RHS proves LHS² = RHS², not LHS = RHS
✅ Correct: Never square both sides in a trig identity verification. Work on one side only.

Worked Examples — 10 Full Identity Verifications

1. Verify: sin²θ(1 + cot²θ) = 1

Apply Pythagorean: 1 + cot²θ = csc²θ
sin²θ · csc²θ = sin²θ · (1/sin²θ) = 1
✓ Identity confirmed

2. Verify: (secθ − 1)(secθ + 1) = tan²θ

Difference of squares: sec²θ − 1
Pythagorean: sec²θ − 1 = tan²θ
✓ Identity confirmed

3. Verify: cosθ/(1 − sinθ) = (1 + sinθ)/cosθ

Cross-multiply (for checking only): cos²θ = (1−sinθ)(1+sinθ) = 1 − sin²θ = cos²θ ✓
Proof approach: LHS × (1+sinθ)/(1+sinθ) = cosθ(1+sinθ)/(1−sin²θ) = cosθ(1+sinθ)/cos²θ = (1+sinθ)/cosθ
✓ Identity confirmed

4. Verify: sin(2θ)/(2cos θ) = sinθ

Replace sin(2θ) = 2sinθcosθ: 2sinθcosθ/(2cosθ)
Cancel 2cosθ: sinθ
✓ Identity confirmed

5. Verify: (1 − cos(2θ))/(sin(2θ)) = tanθ

Substitute: 1−cos(2θ) = 2sin²θ and sin(2θ) = 2sinθcosθ
2sin²θ/(2sinθcosθ) = sinθ/cosθ = tanθ
✓ Identity confirmed

6. Verify: cosθ·secθ = 1

secθ = 1/cosθ, so cosθ·(1/cosθ) = 1
✓ Identity confirmed (reciprocal identity)

7. Verify: cot²θ + 1 = csc²θ

This is the third Pythagorean identity — derived by dividing sin²+cos²=1 by sin²θ
✓ Identity confirmed

8. Verify: sin⁴θ − cos⁴θ = sin²θ − cos²θ

Factor LHS: (sin²θ − cos²θ)(sin²θ + cos²θ)
Apply Pythagorean: sin²θ + cos²θ = 1
(sin²θ − cos²θ)(1) = sin²θ − cos²θ
✓ Identity confirmed

9. Verify: tanθ·sinθ + cosθ = secθ

Replace tanθ = sinθ/cosθ: (sin²θ/cosθ) + cosθ
Common denominator: (sin²θ + cos²θ)/cosθ = 1/cosθ = secθ
✓ Identity confirmed

10. Verify: sin(3θ) = 3sinθ − 4sin³θ

sin(3θ) = sin(2θ+θ) = sin(2θ)cosθ + cos(2θ)sinθ
= (2sinθcosθ)cosθ + (1−2sin²θ)sinθ
= 2sinθcos²θ + sinθ − 2sin³θ
= 2sinθ(1−sin²θ) + sinθ − 2sin³θ = 3sinθ − 4sin³θ
✓ Identity confirmed

Frequently Asked Questions

What is a trig identity?

A trigonometric identity is an equation involving trig functions that is true for ALL values of the variable (for which both sides are defined). Unlike a trig equation (which is true only for specific angles), a trig identity holds universally. Example: sin²θ + cos²θ = 1 is true for every angle θ.

What are the three Pythagorean trig identities?

The three Pythagorean trig identities are: (1) sin²θ + cos²θ = 1 — the fundamental identity from the unit circle; (2) tan²θ + 1 = sec²θ — divide the first by cos²θ; (3) 1 + cot²θ = csc²θ — divide the first by sin²θ. These three Pythagorean identities are the foundation of almost every trig proof.

How do you verify a trig identity?

To verify a trig identity: (1) Work on ONE side only. (2) Start with the more complex side. (3) Convert to sin and cos. (4) Apply Pythagorean substitutions. (5) Simplify until both sides match. Never move terms across the equals sign — that is not a valid verification technique.

What is the difference between a trig identity and a trig equation?

A trig identity is true for ALL values — e.g., sin²θ + cos²θ = 1. A trig equation is true for SPECIFIC values — e.g., sin(x) = 0.5 is only true for x = 30° and x = 150° (in [0°, 360°)). The trig equation solver finds those specific solutions; the identity verifier checks if the equation holds for all values.

How do you solve trig equations?

Use the reference-angle method: (1) Isolate the trig function. (2) Find the reference angle using the inverse. (3) Use ASTC to find which quadrants apply. (4) List all solutions in [0°, 360°). (5) Write the general solution with +360°n (or +180°n for tan). Our trig equation solver does all five steps automatically.

What are the double angle formulas?

sin(2θ) = 2sinθcosθ; cos(2θ) = cos²θ−sin²θ = 2cos²θ−1 = 1−2sin²θ; tan(2θ) = 2tanθ/(1−tan²θ). All three are derived from the sum formulas with A = B = θ.

What is the half angle formula for sin?

sin(θ/2) = ±√((1−cosθ)/2). The sign (±) depends on which quadrant θ/2 falls in — positive in Q1 or Q2, negative in Q3 or Q4. This formula is derived from cos(2α) = 1−2sin²α by substituting α = θ/2.

Related Calculators

Pythagorean Identities

sin²θ + cos²θ = 1 Unit circle — most fundamental

tan²θ + 1 = sec²θ ÷ cos²θ from identity 1

1 + cot²θ = csc²θ ÷ sin²θ from identity 1

Double Angle

sin(2θ) = 2sinθcosθ Sum formula: A=B=θ

cos(2θ) = cos²θ−sin²θ Form 1

cos(2θ) = 1−2sin²θ Form 3 — power reducing

tan(2θ) = 2tanθ/(1−tan²θ) From tan sum formula

Quick Verify