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Learn · Mathematics · Trigonometry

Trigonometric identities and simplifying

The Pythagorean identity and its two rearrangements, tan and cot as quotients, the square identities, and how to simplify a fraction of sines and cosines.

High school Lesson 4 of 9 Practice

First, the rules

Six identities that solve almost everything

You do not need a hundred formulas. You need these, and the habit of rewriting tan and cot in terms of sin and cos.

★ The Pythagorean identity
Examples 1, 3
sin²x + cos²x = 1 ⇒ 1 − sin²x = cos²x, 1 − cos²x = sin²x
★ Quotients
Examples 1, 2, 3
tan x = sin xcos x, cot x = cos xsin x, tan x · cot x = 1
Square identities
Example 1
1 + tan²x = 1cos²x, 1 + cot²x = 1sin²x
Difference of squares (algebra, not trig)
Examples 1, 3
a² − b² = (a − b)(a + b)
Square of a sum or difference
Example 3
(a ± b)² = a² ± 2ab + b²
Where the identity comes from
On the unit circle P = (cos x, sin x) is at distance 1 from the origin: Pythagoras gives cos²x + sin²x = 1. Divide by cos²x for the first square identity, by sin²x for the second.

Method

How to simplify, every time

  1. See 1 − sin² or 1 − cos²? Replace it at once with cos² or sin².
  2. See tan or cot? Write it as a fraction of sin and cos, then put everything over a common denominator.
  3. See a product like (1 − cos x)(1 + cos x)? That is a difference of squares: 1 − cos²x = sin²x.
  4. See a square of a sum or difference? Expand it; the cross term tan x · cot x is just 1.
  5. Cancel common factors top and bottom, and read off the answer.

Step by step

Worked examples

1

Six simplifications

identities

Simplify each expression.

Ask yourself: Whenever you see 1 − sin² or 1 − cos² in a numerator or denominator, what do you write?

Show the solution
  1. a) 1 − sin²xcos x = cos²xcos x = cos x — replace 1 − sin²x, cancel one cos.
  2. b) cos²xsin²x − 1 = cos²x−cos²x = −1 — sin²x − 1 is the negative of 1 − sin²x.
  3. c) (1 − cos 20°)(1 + cos 20°)1 − cos 20° = 1 + cos 20° — the numerator is sin²20° written as a difference of squares; cancel.
  4. d) sin x · cos xsin x = cos x — cot x · sin x: write cot as cos / sin, the sin cancels.
  5. e) tan x + cot x = sin²x + cos²xsin x cos x = 1sin x cos x — common denominator; the numerator is 1.
  6. f) 1 / cos²x1 / sin²x = sin²xcos²x = tan²x — this is (1 + tan²x) / (1 + cot²x); flip and multiply.

Answer cos x, −1, 1 + cos 20°, cos x, 1sin x cos x, tan²x

2

From a ratio equation to a tangent

tan = sin / cos

If sin x + cos x2 sin x − cos x = 34, find tan x.

Ask yourself: Cross-multiply, collect sines on one side, then divide by cos x.

Show the solution
  1. 4 sin x + 4 cos x = 6 sin x − 3 cos x
  2. 7 cos x = 2 sin x
  3. Divide both sides by 2 cos x: sin xcos x = 72

Answer tan x = 72

3

tan²x − cot²x from tan x − cot x

squaring · difference of squares

Given tan x − cot x = √5, find tan²x − cot²x.

Ask yourself: a² − b² = (a − b)(a + b). You have (a − b); how do you get (a + b)?

Show the solution
  1. Square the given: tan²x − 2 tan x cot x + cot²x = 5.
  2. tan · cot = 1, so tan²x + cot²x − 2 = 5, i.e. tan²x + cot²x = 7.
  3. Now the square of the sum: (tan x + cot x)² = 7 + 2 = 9, so tan x + cot x = 3 (positive for an acute x).
  4. tan²x − cot²x = (tan x − cot x)(tan x + cot x) = √5 · 3

Answer 3√5

Practice: simplify

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An expression built from the identities above — choose what it simplifies to. Wrong answers come with the rewrite that gets there. Your best score is kept in this browser only.

About this lesson

The identities every high-school exam uses: sin²x + cos²x = 1 and its rearrangements, tan x = sin x / cos x, cot x = cos x / sin x, tan x · cot x = 1, 1 + tan²x = 1 / cos²x and 1 + cot²x = 1 / sin²x. Worked examples show how to simplify expressions, turn a ratio equation into a tangent, and square a difference to find a sum. Practice questions check your simplification.

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