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Trigonometric Identities

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Core Definition

A trigonometric identity is an equality involving trigonometric functions that is true for every single angle θ\theta in the shared domain of the expressions.

Unlike conditional equations (which are true only for specific values of xx), identities express invariant geometric relationships between circular angles, arc lengths, and right triangle proportions.

Conceptual & Geometric Intuition

All fundamental trigonometric identities originate from the geometry of the Unit Circle (a circle of radius r=1r = 1 centered at the origin (0,0)(0, 0)).

For an angle θ\theta in standard position, the terminal ray intersects the unit circle at the coordinate point:

(x,y)=(cos⁡θ,sin⁡θ)(x, y) = (\cos \theta, \sin \theta)

                     y ^
                       |    (cos theta, sin theta)
                       |     *
                       |    /|
                       |   / |
                   r=1 |  /  | y = sin theta
                       | /   |
                       |/____|________> x
                      O   x = cos theta

1. Pythagorean Identities

The Fundamental Identity

Applying the Pythagorean theorem (x2+y2=r2x^2 + y^2 = r^2) to the right triangle inside the unit circle (r=1r = 1) yields:

cos⁡2θ+sin⁡2θ=1\cos^2 \theta + \sin^2 \theta = 1

  • Geometric Meaning: No matter the angle θ\theta, the sum of the horizontal component squared and the vertical component squared always equals the radius squared (11).
  • Rearrangements for Calculus:
    • sin⁡2θ=1−cos⁡2θ=(1−cos⁡θ)(1+cos⁡θ)\sin^2 \theta = 1 - \cos^2 \theta = (1 - \cos \theta)(1 + \cos \theta)
    • cos⁡2θ=1−sin⁡2θ=(1−sin⁡θ)(1+sin⁡θ)\cos^2 \theta = 1 - \sin^2 \theta = (1 - \sin \theta)(1 + \sin \theta)

Secondary Pythagorean Identities

Dividing sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 by cos⁡2θ\cos^2 \theta or sin⁡2θ\sin^2 \theta generates two additional forms:

  1. Tangent-Secant Identity (divide by cos⁡2θ\cos^2 \theta):
    tan⁡2θ+1=sec⁡2θ  ⟺  sec⁡2θ−1=tan⁡2θ\tan^2 \theta + 1 = \sec^2 \theta \iff \sec^2 \theta - 1 = \tan^2 \theta
  2. Cotangent-Cosecant Identity (divide by sin⁡2θ\sin^2 \theta):
    1+cot⁡2θ=csc⁡2θ  ⟺  csc⁡2θ−1=cot⁡2θ1 + \cot^2 \theta = \csc^2 \theta \iff \csc^2 \theta - 1 = \cot^2 \theta

2. Reciprocal and Quotient Identities

  • Quotient Identities: The tangent represents the slope (m=ΔyΔx=yxm = \frac{\Delta y}{\Delta x} = \frac{y}{x}) of the terminal ray:
    tan⁡θ=sin⁡θcos⁡θandcot⁡θ=cos⁡θsin⁡θ=1tan⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta}
  • Reciprocal Identities:
    sec⁡θ=1cos⁡θ,csc⁡θ=1sin⁡θ,cot⁡θ=1tan⁡θ\sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta}, \quad \cot \theta = \frac{1}{\tan \theta}

3. Even/Odd Symmetry Identities

  • Cosine is Even (Symmetric across y-axis): Reflecting an angle to −θ-\theta keeps the horizontal coordinate xx identical:
    cos⁡(−θ)=cos⁡θandsec⁡(−θ)=sec⁡θ\cos(-\theta) = \cos \theta \quad \text{and} \quad \sec(-\theta) = \sec \theta
  • Sine and Tangent are Odd (Origin Symmetry): Reflecting to −θ-\theta flips the vertical coordinate yy:
    sin⁡(−θ)=−sin⁡θandtan⁡(−θ)=−tan⁡θ\sin(-\theta) = -\sin \theta \quad \text{and} \quad \tan(-\theta) = -\tan \theta

4. Double-Angle Identities

Derived from the angle addition formulas (sin⁡(α+β)\sin(\alpha + \beta) and cos⁡(α+β)\cos(\alpha + \beta) with α=β=θ\alpha = \beta = \theta):

  • Sine Double-Angle: sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin \theta \cos \theta
  • Cosine Double-Angle:
    cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta

5. Trigonometric Conjugate Pairs

In Calculus limits and integrals, expressions like (1−cos⁡θ)(1 - \cos \theta) or (1−sin⁡θ)(1 - \sin \theta) frequently cause division by zero. Multiplying by their trigonometric conjugate converts them into single squared terms via difference of squares:

(1−cos⁡θ)(1+cos⁡θ)=1−cos⁡2θ=sin⁡2θ(1 - \cos \theta)(1 + \cos \theta) = 1 - \cos^2 \theta = \sin^2 \theta
(1−sin⁡θ)(1+sin⁡θ)=1−sin⁡2θ=cos⁡2θ(1 - \sin \theta)(1 + \sin \theta) = 1 - \sin^2 \theta = \cos^2 \theta

Example Problems

Example Problem 1: Simplifying Difference of Squares

Simplify the expression:

1−sin⁡2xcos⁡x\frac{1 - \sin^2 x}{\cos x}

Solution

Step 1: Apply the Pythagorean identity
Substitute 1−sin⁡2x=cos⁡2x1 - \sin^2 x = \cos^2 x:

cos⁡2xcos⁡x\frac{\cos^2 x}{\cos x}

Step 2: Cancel common factor

cos⁡2xcos⁡x=cos⁡x(for cos⁡x≠0)\frac{\cos^2 x}{\cos x} = \cos x \quad (\text{for } \cos x \neq 0)

Example Problem 2: Transforming with Trigonometric Conjugates

Multiply (1−cos⁡θ)(1 - \cos \theta) by its conjugate and rewrite the result in terms of sin⁡θ\sin \theta.

Solution

Step 1: Multiply by conjugate (1+cos⁡θ)(1 + \cos \theta)

(1−cos⁡θ)(1+cos⁡θ)=12−cos⁡2θ=1−cos⁡2θ(1 - \cos \theta)(1 + \cos \theta) = 1^2 - \cos^2 \theta = 1 - \cos^2 \theta

Step 2: Apply Pythagorean identity

1−cos⁡2θ=sin⁡2θ1 - \cos^2 \theta = \sin^2 \theta

Answer: sin⁡2θ\sin^2 \theta

Connections & References