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Trigonometric Identities — Definition, Formula & Examples

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides are defined. They serve as foundational tools for simplifying expressions, solving equations, and proving other mathematical results.

A trigonometric identity is a statement of equality between two expressions composed of trigonometric functions (sin, cos, tan, csc, sec, cot) and their arguments, which holds for all values in the domain common to both sides. The fundamental set includes the Pythagorean identities, ratio identities, reciprocal identities, and angle-transformation identities (sum/difference, double angle, half angle).

Key Formula

\sin^2\theta + \cos^2\theta = 1$$ $$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta
Where:
  • θ\theta = Any angle (in degrees or radians) where the functions are defined

How It Works

Trigonometric identities let you rewrite one trig expression as another equivalent form. To simplify a complex expression, you substitute known identities until the expression reduces. To prove an identity, you typically work on one side—transforming it step by step until it matches the other side. The three Pythagorean identities (sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 and its two variants) are the most frequently used starting points. Ratio and reciprocal identities convert between the six trig functions, while sum/difference and double-angle identities handle compound angles.

Worked Example

Problem: Simplify the expression sin⁡2θ1−cos⁡θ\dfrac{\sin^2\theta}{1 - \cos\theta}.
Step 1: Use the Pythagorean identity to replace sin⁡2θ\sin^2\theta.
sin⁡2θ=1−cos⁡2θ\sin^2\theta = 1 - \cos^2\theta
Step 2: Substitute into the original expression and factor the numerator as a difference of squares.
1−cos⁡2θ1−cos⁡θ=(1−cos⁡θ)(1+cos⁡θ)1−cos⁡θ\frac{1 - \cos^2\theta}{1 - \cos\theta} = \frac{(1 - \cos\theta)(1 + \cos\theta)}{1 - \cos\theta}
Step 3: Cancel the common factor (1−cos⁡θ)(1 - \cos\theta), valid when cos⁡θ≠1\cos\theta \neq 1.
=1+cos⁡θ= 1 + \cos\theta
Answer: sin⁡2θ1−cos⁡θ=1+cos⁡θ\dfrac{\sin^2\theta}{1 - \cos\theta} = 1 + \cos\theta (for cos⁡θ≠1\cos\theta \neq 1).

Another Example

Problem: Prove that tan⁡θ+cot⁡θ=sec⁡θ csc⁡θ\tan\theta + \cot\theta = \sec\theta\,\csc\theta.
Step 1: Rewrite the left side using ratio identities.
tan⁡θ+cot⁡θ=sin⁡θcos⁡θ+cos⁡θsin⁡θ\tan\theta + \cot\theta = \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta}
Step 2: Combine over a common denominator.
=sin⁡2θ+cos⁡2θcos⁡θ sin⁡θ= \frac{\sin^2\theta + \cos^2\theta}{\cos\theta\,\sin\theta}
Step 3: Apply the Pythagorean identity: sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1.
=1cos⁡θ sin⁡θ=sec⁡θ csc⁡θ= \frac{1}{\cos\theta\,\sin\theta} = \sec\theta\,\csc\theta
Answer: Both sides equal sec⁡θ csc⁡θ\sec\theta\,\csc\theta, so the identity is proven.

Why It Matters

Trigonometric identities are central to precalculus and AP Calculus, where you need them to evaluate integrals like ∫sin⁡2x dx\int \sin^2 x\, dx and to simplify derivatives. In physics, they appear whenever you decompose forces, analyze waves, or work with oscillations. Engineers use them in signal processing to convert between product and sum forms of waveforms.

Common Mistakes

Mistake: Writing sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 but then incorrectly claiming sin⁡θ+cos⁡θ=1\sin\theta + \cos\theta = 1.
Correction: The identity applies to the squares of sine and cosine, not to the functions themselves. sin⁡θ+cos⁡θ\sin\theta + \cos\theta does not simplify to 1.
Mistake: Manipulating both sides of an identity simultaneously when trying to prove it.
Correction: In a proof, work on only one side and transform it into the other. Working on both sides at once can introduce invalid steps and circular reasoning.

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