Fundamental Identities
Derive and apply the reciprocal identities, quotient identities, and the three Pythagorean identities (sin²θ + cos²θ = 1 and its two variations). Use these as the basis for all identity verification work.
A lesson of Pre-Calculus, in Trigonometric Identities and Equations, about 10 minutes.
Fundamental Trigonometric Identities
By the end of this lesson, you will be able to state and apply all fundamental trig identities (reciprocal, quotient, Pythagorean, cofunction, and even/odd), simplify trig expressions using identities, and recognize which identities are most useful in a given context.
The Big Idea
Identities are equations true for all values in their domains — not just specific solutions. Unlike trig equations (which have specific solutions), identities are algebraic tools. Mastering them unlocks the ability to simplify, evaluate, and transform any trig expression — essential for calculus.
The Identity Groups
Reciprocal Identities
$$\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}$$
Quotient Identities
$$\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}$$
Pythagorean Identities
$$\sin^2\theta + \cos^2\theta = 1$$ $$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta$$
Derived by dividing the first by $\cos^2\theta$ or $\sin^2\theta$.
Cofunction Identities
$$\sin\theta = \cos\left(\frac{\pi}{2} - \theta\right), \quad \cos\theta = \sin\left(\frac{\pi}{2} - \theta\right)$$ $$\tan\theta = \cot\left(\frac{\pi}{2} - \theta\right), \quad \sec\theta = \csc\left(\frac{\pi}{2} - \theta\right)$$
Even/Odd Identities
Even (cosine, secant): $\cos(-\theta) = \cos\theta$; $\sec(-\theta) = \sec\theta$
Odd (sine, cosecant, tangent, cotangent): $\sin(-\theta) = -\sin\theta$; $\tan(-\theta) = -\tan\theta$
Simplifying Trig Expressions
Strategy: convert everything to sines and cosines, then simplify algebraically, or use a Pythagorean identity to substitute.
Example 1 — Basic Simplification
Simplify: $\dfrac{\sin^2\theta}{\cos\theta} + \cos\theta$
$$= \frac{\sin^2\theta + \cos^2\theta}{\cos\theta} = \frac{1}{\cos\theta} = \sec\theta$$
Example 2 — Using Pythagorean Identity
Simplify: $\sec^2\theta - \tan^2\theta$
By the identity $1 + \tan^2\theta = \sec^2\theta$:
$\sec^2\theta - \tan^2\theta = 1$