Trigonometric Identities Grid- Complete Reference

What This Grid Actually Is

This is a complete reference for trigonometric identities — every formula you need, organized by category. No proofs, no derivations, just the identities themselves.

If you are cramming for an exam or solving problems, bookmark this page. It has everything.

🔺 The Pythagorean Identities

These are the foundation. Everything else branches from here.

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

The first one is the most important. Memorize it first.

↔️ Reciprocal Identities

These define the basic relationships between the six trig functions.

• sin θ = 1/csc θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ
• csc θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ

÷ The Quotient Identities

Tan and cot defined as ratios of sin and cos.

• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

🔄 Co-Function Identities

These relate functions of complementary angles (angles that add to 90° or π/2).

• sin(π/2 - θ) = cos θ
• cos(π/2 - θ) = sin θ
• tan(π/2 - θ) = cot θ
• cot(π/2 - θ) = tan θ
• sec(π/2 - θ) = csc θ
• csc(π/2 - θ) = sec θ

⁻ Even and Odd Identities

These describe symmetry. Know which functions are even and which are odd.

• cos(-θ) = cos θ — cosine is even
• sec(-θ) = sec θ — secant is even
• sin(-θ) = -sin θ — sine is odd
• tan(-θ) = -tan θ — tangent is odd
• csc(-θ) = -csc θ — cosecant is odd
• cot(-θ) = -cot θ — cotangent is odd

×2 Double Angle Identities

Express functions of 2θ in terms of single angles.

Sine Double Angle

• sin(2θ) = 2 sin θ cos θ

Cosine Double Angle

• cos(2θ) = cos²θ - sin²θ
• cos(2θ) = 2cos²θ - 1
• cos(2θ) = 1 - 2sin²θ

Tangent Double Angle

• tan(2θ) = 2tan θ / (1 - tan²θ)

÷2 Half Angle Identities

Use these when you need functions of θ/2.

• sin(θ/2) = ±√((1 - cos θ) / 2)
• cos(θ/2) = ±√((1 + cos θ) / 2)
• tan(θ/2) = (1 - cos θ) / sin θ
• tan(θ/2) = sin θ / (1 + cos θ)

The ± sign depends on which quadrant the angle falls in.

± Sum and Difference Identities

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)

×→+ Product-to-Sum Identities

Convert products into sums. Useful in integration and signal processing.

• sin A cos B = ½[sin(A + B) + sin(A - B)]
• cos A sin B = ½[sin(A + B) - sin(A - B)]
• cos A cos B = ½[cos(A + B) + cos(A - B)]
• sin A sin B = ½[cos(A - B) - cos(A + B)]

+→× Sum-to-Product Identities

The reverse of product-to-sum.

• sin A + sin B = 2 sin((A + B)/2) cos((A - B)/2)
• sin A - sin B = 2 cos((A + B)/2) sin((A - B)/2)
• cos A + cos B = 2 cos((A + B)/2) cos((A - B)/2)
• cos A - cos B = -2 sin((A + B)/2) sin((A - B)/2)

📐 Law of Sines and Cosines

These solve triangles. Not technically identities, but they use trig functions.

Law of Sines

a/sin A = b/sin B = c/sin C

Law of Cosines

c² = a² + b² - 2ab cos C

Use this when you have two sides and the included angle, or all three sides.

🔄 Power Reducing Identities

Eliminate exponents on powers of sine and cosine.

• sin²θ = (1 - cos(2θ)) / 2
• cos²θ = (1 + cos(2θ)) / 2
• sin³θ = (3 sin θ - sin(3θ)) / 4
• cos³θ = (3 cos θ + cos(3θ)) / 4

📊 Quick Reference Table

⚠️ Common Mistakes to Avoid

• Confusing sin(2θ) with 2 sin θ — they are not equal
• Forgetting the ± in half-angle formulas — check the quadrant
• Mixing up sum/difference formulas — signs matter
• Using the wrong Pythagorean identity — pick the one that matches your given values
• Forgetting that tan, cot, sec, csc have restrictions (undefined at certain angles)

🛠️ How to Use This Reference

Here is how to approach identity problems:

1. Identify what you know and what you need. Write down the given expression or equation.
2. Match to an identity. Scan this list for a formula that contains your known values.
3. Start with Pythagorean identities if you see sin² + cos² or 1 - sin² patterns.
4. Use sum/difference formulas when splitting angles or combining functions.
5. Convert everything to sin and cos if you are stuck — the reciprocal identities make this easy.
6. Simplify step by step. Do not try to skip from the start to the finish in one jump.

Example: Simplify (1 - cos²θ)/sin θ

Step 1: Replace 1 - cos²θ using sin²θ + cos²θ = 1

1 - cos²θ = sin²θ

Step 2: Now you have sin²θ / sin θ

Step 3: Simplify to sin θ

That is it. Three steps.

Example: Find sin(75°) Using Sum Identities

Break 75° into angles you know: 75° = 45° + 30°

sin(75°) = sin(45° + 30°)

= sin 45° cos 30° + cos 45° sin 30°

= (√2/2)(√3/2) + (√2/2)(1/2)

= √6/4 + √2/4

= (√6 + √2)/4

That is how you apply these identities in practice.

What to Memorize

You do not need all of these. Memorize these first:

• sin²θ + cos²θ = 1
• sin(2θ) = 2 sin θ cos θ
• cos(2θ) = cos²θ - sin²θ
• sin(A ± B) and cos(A ± B) formulas
• tan(A + B) formula
• Law of Sines and Cosines

The rest you can derive if you know these well. The relationships between them matter more than rote memorization.