Learn · Mathematics · Class 10

Trigonometric Identities

The relationships between the ratios, and proving one side equals the other.

11 parts
Scroll to begin ↓
📘 DefinitionIdentity versus equationPart 1 of 11
🧩 The partsThe fundamental identitiesPart 2 of 11
🔍 In detailWhy sin²θ + cos²θ = 1Part 3 of 11
✏️ Worked exampleProving an identityPart 4 of 11
✏️ Worked exampleA harder proofPart 5 of 11
Which ratios are positive whereAallSsinTtanCcos0°90°180°270°CASTRead anticlockwise fromthe fourth quadrantOnly the named ratiois positive theresin²θ + cos²θ = 1 — the identity everything follows from
Only the named ratio is positive in each quadrant.
✏️ Worked exampleSolving a trigonometric equationPart 7 of 11
🌍 In the real worldWhere identities are usedPart 8 of 11
⚠️ Common mistakesWhere marks are lostPart 9 of 11
✅ SummaryWhat to rememberPart 10 of 11
✏️ Worked exampleWatch one worked throughPart 11 of 11
1📘 DefinitionPart 1 / 11

Identity versus equation

An identity is true for every value of the variable. An equation is true only for particular values.

sin²θ + cos²θ = 1  , identity, always true
sin θ = 0.5  , equation, true for specific θ

The difference in what you do: you prove an identity and solve an equation.

Note the notation: sin²θ means (sin θ)², not sin(θ²). The two are entirely different.

2🧩 The partsPart 2 / 11

The fundamental identities

The Pythagorean identity: everything else follows from it:

sin²θ + cos²θ = 1

Rearranged forms, both used constantly:

sin²θ = 1 − cos²θ
cos²θ = 1 − sin²θ

The quotient identity:

tan θ = sin θ / cos θ

The reciprocal identities:

cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ = cos θ / sin θ
3🔍 In detailPart 3 / 11

Why sin²θ + cos²θ = 1

This is Pythagoras, rewritten, which is worth seeing rather than memorising.

Take a right-angled triangle with hypotenuse h, opposite o and adjacent a.

sin θ = o/h    cos θ = a/h

Square and add:

sin²θ + cos²θ = o²/h² + a²/h²
= (o² + a²) / h²

But Pythagoras says o² + a² = h². So:

= h²/h² = 1

Why this matters: a result you can derive is one you can rebuild under pressure. A result you only memorised disappears when memory fails.

4✏️ Worked examplePart 4 / 11

Proving an identity

Prove: (1 − cos²θ) / sin θ = sin θ

The rules of proof, these carry marks:

  • Work on one side only, usually the more complicated one.
  • Transform it until it becomes the other side.
  • Never move terms across the equals sign, that assumes what you are proving.

Start with the left-hand side:

LHS = (1 − cos²θ) / sin θ

Use the Pythagorean identity, the numerator is sin²θ:

= sin²θ / sin θ

Cancel:

= sin θ = RHS  ∎

The reflex to build: whenever you see 1 − cos²θ or 1 − sin²θ, replace it immediately. That substitution unlocks most proofs at this level.

5✏️ Worked examplePart 5 / 11

A harder proof

Prove: tan θ + cot θ = 1 / (sin θ cos θ)

Step 1: convert everything to sin and cos. This is almost always the right first move.

LHS = sin θ/cos θ + cos θ/sin θ

Step 2: common denominator.

= (sin²θ + cos²θ) / (sin θ cos θ)

Step 3: apply the Pythagorean identity. The numerator is 1.

= 1 / (sin θ cos θ) = RHS  ∎

The general strategy for any identity proof:

  1. Convert to sin and cos.
  2. Combine over a common denominator.
  3. Look for sin²θ + cos²θ to replace with 1.
  4. Simplify.

That sequence handles the great majority of proofs set at this level.

6🔍 In detailPart 6 / 11

Signs in each quadrant

Beyond 90°, the ratios take negative values in a predictable pattern.

CAST, read anticlockwise starting from the fourth quadrant, names the ratio that is positive in each:

  • 1st (0-90°), All positive.
  • 2nd (90-180°): only Sin positive.
  • 3rd (180-270°): only Tan positive.
  • 4th (270-360°): only Cos positive.

Why this matters: a calculator returns only one answer for sin⁻¹. If the question asks for all solutions in 0° to 360°, there is usually a second one, and CAST tells you which quadrant it is in.

7✏️ Worked examplePart 7 / 11

Solving a trigonometric equation

Question. Solve 2 sin θ = 1 for 0° ≤ θ ≤ 360°.

Step 1: isolate the ratio.

sin θ = 0.5

Step 2: the principal value from the calculator.

θ = sin⁻¹(0.5) = 30°

Step 3: find the other solutions. Sin is positive in the first and second quadrants, so there is a second answer.

second solution = 180° − 30° = 150°

Answer: θ = 30° or 150°

Check both: sin 150° = 0.5 ✅

The single most common error in this topic is stopping at 30°. The range given in the question (0° to 360°) is the signal that more than one solution is expected. Always ask how many the range allows.

8🌍 In the real worldPart 8 / 11

Where identities are used

Simplifying before solving. A complicated trigonometric expression often reduces to something manageable, and identities are how.

Physics. Resolving forces and analysing oscillations produce expressions that only simplify through identities.

Waves. Sound, light and alternating current are described by sine functions; combining them requires identities.

Engineering and signal processing. Converting between forms of a wave equation is routine identity work.

In the examination: proofs are predictable in structure. Practising the four-step strategy makes them among the more reliable marks available.

9⚠️ Common mistakesPart 9 / 11

Where marks are lost

  • sin²θ read as sin(θ²).
  • Working on both sides of an identity, or moving terms across.
  • Giving only the calculator value when the range allows more.
  • Calculator in radians.
  • Cancelling sin θ from both sides of an equation. That loses the solutions where sin θ = 0.
  • Forgetting CAST and giving a wrongly signed answer.
10✅ SummaryPart 10 / 11

What to remember

  • sin²θ + cos²θ = 1: it is Pythagoras rewritten.
  • Proofs: one side only; convert to sin and cos first.
  • CAST gives the signs and the second solution.
  • Check how many solutions the range allows.
11✏️ Worked examplePart 11 / 11

Watch one worked through

Here is one done slowly, with the reason for every move. Then one for you.

Watch one first

Prove that (1 - cos²θ) ÷ sinθ = sinθ.

  1. Start from the more complicated side. Here that is the left.
    LHS = (1 − cos²θ) ÷ sinθ
    Always work one side down to the other. Doing things to both sides at once is not a proof and loses marks.
  2. Use the Pythagorean identity to replace the numerator.
    sin²θ + cos²θ = 1 → 1 − cos²θ = sin²θ
    This one identity, rearranged three ways, solves most proofs at this level. Learn all three arrangements.
  3. Substitute it in.
    LHS = sin²θ ÷ sinθ
    Replacing rather than cancelling keeps every line a true statement, which is what a proof has to be.
  4. Cancel one sinθ and state the conclusion.
    = sinθ = RHS ∎
    Write that it equals the right-hand side. An examiner wants to see you noticed you had finished.

Answer: LHS becomes sin²θ ÷ sinθ which is sinθ, equal to the RHS

Now you try

What does 1 - sin²θ equal? Write it as cos squared theta in the form cos^2.

Show the answer

cos^2

Practise Trigonometric Identities

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🧠 5 quizzes on Trigonometric Identities (2 easy, 2 medium, 1 hard)

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🖨️ Printable worksheet

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