In this lesson you’ll learn the fundamental trigonometric identities and how to use them to simplify expressions and solve problems.
Trigonometric identities are equations involving trig functions that are true for all values of the variable (where defined). They’re not formulas you solve. They’re relationships that are always true, and you use them to rewrite expressions in simpler or more useful forms.
sin θ = 1 csc θ cos θ = 1 sec θ tan θ = 1 cot θ \sin\theta = \frac{1}{\csc\theta} \qquad \cos\theta = \frac{1}{\sec\theta} \qquad \tan\theta = \frac{1}{\cot\theta} sin θ = csc θ 1 cos θ = sec θ 1 tan θ = cot θ 1 tan θ = sin θ cos θ cot θ = cos θ sin θ \tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta} tan θ = cos θ sin θ cot θ = sin θ cos θ These are the most important ones. They come directly from the Pythagorean theorem applied to the unit circle.
sin 2 θ + cos 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ 1 + \tan^2\theta = \sec^2\theta 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ 1 + \cot^2\theta = \csc^2\theta 1 + cot 2 θ = csc 2 θ The first one is the big one. The other two are derived from it by dividing through by cos²θ or sin²θ.
You can also rearrange them:
sin 2 θ = 1 − cos 2 θ \sin^2\theta = 1 - \cos^2\theta sin 2 θ = 1 − cos 2 θ cos 2 θ = 1 − sin 2 θ \cos^2\theta = 1 - \sin^2\theta cos 2 θ = 1 − sin 2 θ tan 2 θ = sec 2 θ − 1 \tan^2\theta = \sec^2\theta - 1 tan 2 θ = sec 2 θ − 1
1. Simplify sin θ cos θ + cos θ sin θ \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} c o s θ s i n θ + s i n θ c o s θ
Recognize the quotient identities: this is tan θ + cot θ \tan\theta + \cot\theta tan θ + cot θ
To combine into one fraction, find a common denominator:
sin 2 θ + cos 2 θ sin θ cos θ = 1 sin θ cos θ \frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta} = \frac{1}{\sin\theta\cos\theta} sin θ cos θ sin 2 θ + cos 2 θ = sin θ cos θ 1 By the Pythagorean identity, the numerator sin 2 θ + cos 2 θ \sin^2\theta + \cos^2\theta sin 2 θ + cos 2 θ
2. Verify sin 2 θ + cos 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1
sin ( π 3 ) = 3 2 cos ( π 3 ) = 1 2 \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \qquad \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} sin ( 3 π ) = 2 3 cos ( 3 π ) = 2 1 ( 3 2 ) 2 + ( 1 2 ) 2 = 3 4 + 1 4 = 1 ✓ \left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1 \quad \checkmark ( 2 3 ) 2 + ( 2 1 ) 2 = 4 3 + 4 1 = 1 ✓ 3. Simplify sec 2 θ − tan 2 θ \sec^2\theta - \tan^2\theta sec 2 θ − tan 2 θ
We know that 1 + tan²θ = sec²θ. Rearranging:
sec 2 θ − tan 2 θ = 1 \sec^2\theta - \tan^2\theta = 1 sec 2 θ − tan 2 θ = 1 That’s it. It always equals 1, no matter what θ is.
Trigonometric identities are used in:
Simplifying complex waveforms in audio and signal processing
Solving equations in physics (wave motion, optics, electricity)
Engineering calculations (mechanical vibrations, structural analysis)
Computer graphics (efficient rotation and transformation calculations)
Example: in electronics, identities help simplify expressions for alternating current and voltage relationships. Instead of working with a complicated expression, you can use an identity to reduce it to something much simpler.
The Pythagorean identity sin²θ + cos²θ = 1 is your most powerful tool. Memorize the reciprocal and quotient identities, then practice using them to rewrite expressions. The goal is to become comfortable transforming trig expressions into simpler forms. This skill is essential for verifying identities and solving trig equations in the next lessons.
Which identity is correct? A. $\sin^2\theta - \cos^2\theta = 1$ B. $\sin^2\theta + \cos^2\theta = 1$ C. $\tan^2\theta + \cot^2\theta = 1$ D. $\sec^2\theta - \csc^2\theta = 1$
$\tan\theta$ equals: A. $\frac{1}{\cos\theta}$ B. $\frac{\cos\theta}{\sin\theta}$ C. $\frac{1}{\sin\theta}$ D. $\frac{\sin\theta}{\cos\theta}$
The reciprocal of cosine is: A. Secant B. Cosecant C. Tangent D. Cotangent
$\sec^2\theta - \tan^2\theta$ equals: A. $-1$ B. $0$ C. $1$ D. $\sin^2\theta$
$\sin^2\theta$ can be rewritten as: A. $1 + \cos^2\theta$ B. $1 - \cos^2\theta$ C. $\cos^2\theta - 1$ D. $\tan^2\theta$
Retry Quiz
