Introduction to Trigonometry · Lesson 4 of 5
Trigonometric Identities
“Different-looking trigonometric expressions discover they have been equal all along.”
• By the end of this lesson, you should be able to. • Explain what a trigonometric identity is. • Derive the three fundamental identities from Pythagoras theorem. • Use identities to express one trigonometric ratio in terms of another. • Prove identities by simplifying one side carefully. • Recognise common strategies and mistakes in identity problems.
An ordinary equation may be true only for particular values. An identity is stronger: it is true for every value for which both sides are defined. Trigonometric identities are equations built from trigonometric ratios that remain true throughout their allowed domain.
Equation vs Identity
| Equation | Identity |
|---|---|
| May be true only for some values | True for all allowed values |
| Example: x + 2 = 5 | Example: sin²A + cos²A = 1 |
| Usually solved for unknown values | Usually proved by transforming expressions |
Deriving the First Identity
Take right triangle ABC, right-angled at B. By Pythagoras theorem, AB² + BC² = AC². Divide every term by AC². Since AB/AC = cos A and BC/AC = sin A, the result becomes the first fundamental trigonometric identity.
This identity is often the starting point when an expression contains sin²A and cos²A. It can be rearranged as sin²A = 1 - cos²A or cos²A = 1 - sin²A.
Deriving the Second Identity
Start again from AB² + BC² = AC², but divide every term by AB². Then BC/AB = tan A and AC/AB = sec A.
tan A and sec A are not defined at 90°, so this identity is used where those ratios are defined.
Deriving the Third Identity
This time divide AB² + BC² = AC² by BC². Then AB/BC = cot A and AC/BC = cosec A.
Using an Identity to Find Other Ratios
Problem
If tan A = 3/4 and A is acute, find sec A, cos A, sin A and cosec A.
- 1.Use 1 + tan²A = sec²A.
- 2.sec²A = 1 + (3/4)² = 1 + 9/16 = 25/16.
- 3.Since A is acute, sec A is positive, so sec A = 5/4.
- 4.Therefore cos A = 1/sec A = 4/5.
- 5.Using sin²A + cos²A = 1: sin²A = 1 - 16/25 = 9/25.
- 6.So sin A = 3/5 and cosec A = 5/3.
Expressing Ratios in Terms of Another Ratio
Problem
Express cos A, tan A and sec A in terms of sin A, assuming A is acute.
- 1.From sin²A + cos²A = 1, cos²A = 1 - sin²A.
- 2.Since A is acute, cos A > 0.
- 3.Therefore cos A = √(1 - sin²A).
- 4.tan A = sin A / cos A = sin A / √(1 - sin²A).
- 5.sec A = 1/cos A = 1 / √(1 - sin²A).
How to Prove a Trigonometric Identity
| Step | What to do |
|---|---|
| 1 | Start with the more complicated side, usually the LHS. |
| 2 | Convert sec, cosec, tan or cot into sin and cos if that makes the expression simpler. |
| 3 | Use one of the three fundamental identities when you see a useful square pattern. |
| 4 | Factor, take a common denominator or rationalise when useful. |
| 5 | Stop as soon as you reach the other side. Do not manipulate both sides independently unless the method is explicitly justified. |
Problem
Prove that sec A(1 - sin A)(sec A + tan A) = 1.
- 1.Start with the LHS.
- 2.sec A(1 - sin A)(sec A + tan A)
- 3.= (1/cos A)(1 - sin A)[1/cos A + sin A/cos A]
- 4.= (1 - sin A)(1 + sin A) / cos²A
- 5.= (1 - sin²A) / cos²A
- 6.= cos²A / cos²A
- 7.= 1, which is the RHS.
Problem
Prove that (cot A - cos A)/(cot A + cos A) = (cosec A - 1)/(cosec A + 1).
- 1.Begin with the LHS.
- 2.Replace cot A by cos A/sin A.
- 3.LHS = [cos A/sin A - cos A] / [cos A/sin A + cos A].
- 4.Factor cos A in numerator and denominator; it cancels.
- 5.LHS = (1/sin A - 1)/(1/sin A + 1).
- 6.Since 1/sin A = cosec A, LHS = (cosec A - 1)/(cosec A + 1) = RHS.
• An identity is not proved by checking one angle such as 30° or 45°. • Do not cancel terms across addition or subtraction. • Do not replace sin²A + cos²A by something other than 1. • Remember sec²A - tan²A = 1, not sec A - tan A = 1. • Watch undefined values when denominators contain sin A or cos A.
Quiz
How is a trigonometric identity different from an ordinary equation?
Which is the first fundamental trigonometric identity?
Which identity is obtained by dividing the Pythagoras relation by the square of the adjacent side?
What is the value of 9sec²A − 9tan²A?
What is usually the best first step when proving a trigonometric identity?
Practice Problems
- Express sin A, sec A and tan A in terms of cot A.
- Write all the other trigonometric ratios of angle A in terms of sec A.
- Choose the correct value and justify: 16 sec²A - 16 tan²A = (A) 1 (B) 16 (C) 15 (D) 0.
- Prove: (cosec θ - cot θ)² = (1 - cos θ)/(1 + cos θ).
- Prove: (1 + sin A)/(cos A) + cos A/(1 + sin A) = 2 sec A.
- Prove: (sin A + cosec A)² + (cos A + sec A)² = 7 + tan²A + cot²A.
Key Takeaways
• A trigonometric identity is true for all allowed values of the angle. • The three fundamental identities come directly from Pythagoras theorem. • sin²A + cos²A = 1. • 1 + tan²A = sec²A. • 1 + cot²A = cosec²A. • In proofs, simplify strategically; converting everything to sin and cos is often useful. • Identities can also help find unknown trigonometric ratios.
Next, we will bring the whole chapter together in a compact revision lesson, solve mixed examples and practise exam-style questions.