Trigonometric Identities and Triangles
The Pythagorean, Reciprocal and Quotient Identities
The one identity everything follows from, the two derived by division, and the reciprocal and quotient relations used to simplify expressions.
What this page covers
- Derive sin2t + cos2t = 1 from the unit circle
- Derive the other two Pythagorean identities by division
- State the reciprocal and quotient identities
- Simplify a trigonometric expression using them
The fundamental identity
The derivation is one line. The point (cos t, sin t) lies on the unit circle, whose equation is x2 + y2 = 1. Substituting gives the identity.
sin2t means (sin t)2, not sin(t2). The exponent is written after the function name purely to avoid a pile of brackets. There is one genuine exception: sin-1t means the inverse function, not 1sin t, and the source flags this explicitly in Week 8.
Rearranged, sin2t = 1 - cos2t and cos2t = 1 - sin2t. The source records both, and they are used constantly to convert an expression into a single function.
The two derived identities
Dividing the fundamental identity by cos2t or sin2t produces two more. The source performs both divisions in full.
Dividing by cos2t
Rearranged: tan2t = sec2t - 1, which the source also records.
Dividing by sin2t
Rearranged: cot2t = csc2t - 1.
| Identity | Obtained by | Rearranged |
|---|---|---|
| sin2t + cos2t = 1 | The unit circle | sin2t = 1 - cos2t |
| tan2t + 1 = sec2t | Dividing by cos2t | tan2t = sec2t - 1 |
| 1 + cot2t = csc2t | Dividing by sin2t | cot2t = csc2t - 1 |
Only the first needs remembering. The other two are two lines of work away, and deriving them takes less time than recovering from misremembering them.
Reciprocal and quotient identities
Week 8, page 6 writes the second quotient identity as cot θ = cos θsin θ in one place and as sin θsin θ in another. The correct form is cos θsin θ, since cotangent is the reciprocal of tangent. The page's own coordinate definition, cot t = xy, confirms it.
These are the parity statements. Reflecting a point in the x-axis keeps its x coordinate and negates its y, so cosine is even and sine is odd. Tangent, as their ratio, is odd. These are used constantly in deriving the subtraction formulae from the addition ones.
Simplifying with the identities
Worked example — the source's first
Simplify 1 - cos2θsin θ.
The whole simplification is one substitution: 1 - cos2θ is sin2θ. Recognising that pattern is worth more than any amount of algebraic manipulation.
The result holds where sin θ ≠ 0. At θ = 0 the original is 00, which is undefined, while sin θ is 0. The restriction belongs with the answer.
Worked example — the source's second
Simplify sin t cos ttan t.
Converting tangent to sin tcos t turned the problem into ordinary fraction arithmetic. That is the general strategy.
- Convert everything to sine and cosine. This removes the four derived functions and leaves ordinary algebra.
- Look for 1 - cos2 or 1 - sin2 and substitute.
- Combine fractions over a common denominator.
- Factor where possible, and cancel.
- Convert back if a compact form in the other functions is wanted.
Using an identity to find other values
Worked example — the source's case
Given sin t = -1213 with -π2 < t < 0, find cos t and tan t.
The identity gives the magnitude; the stated interval supplies the sign. On -π2 < t < 0 the point is in quadrant IV, where cosine is positive.
(1213)2 + (513)2 = 144 + 25169 = 1 ✓. The 5-12-13 triple appearing here is not a coincidence: any point on the unit circle with rational coordinates comes from a Pythagorean triple.
The identity alone never determines the sign. It gives cos2t, so both roots are algebraically available and the quadrant must decide. Omitting the interval makes the question unanswerable.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| sin2t read as sin(t2) | (sin t)2 | The exponent applies to the value |
| sin t + cos t = 1 | It is the squares that sum to 1 | At t = π4 the sum is √2 |
| tan2t + 1 = csc2t | = sec2t | Dividing by cos2 gives secant |
| cot θ = sin θcos θ | = cos θsin θ | Cotangent is the reciprocal of tangent |
| Taking only the positive root | The quadrant decides the sign | State the interval |
| sin-1t = 1sin t | It is the inverse function | Use csc t for the reciprocal |
Frequently asked questions
Where does sin2t + cos2t = 1 come from?
Straight from the unit circle. The point (cos t, sin t) lies at distance 1 from the origin, so x2 + y2 = 1. It is Pythagoras with the hypotenuse set to 1.
Why divide by cos2t?
Because it converts the identity into one relating tangent and secant. Dividing by sin2t instead gives the cotangent-cosecant version. Both are the same identity in different clothes.
What does sin2t mean?
(sin t)2 — the sine taken first, then squared. It does not mean sin(t2). The notation is compact but ambiguous-looking, and the convention has to be learned.
Which identity should I reach for first?
Usually sin2 + cos2 = 1, in whichever direction helps. Replacing 1 - cos2t by sin2t is the single most common simplifying move.
Source. Handwritten teaching notes, Week 7 pages 2 and 4, and Week 8 pages 6-7.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.