Author: Indigo Curnick
Date: 2026-06-06
Quotient Identities
$$\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}$$
$$\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}}$$
Pythagorean Identities
$$\sin^2{\theta} + \cos^2{\theta} = 1$$ $$\sec^2{\theta} - \tan^2{\theta} = 1$$ $$\csc^2{\theta} - \cot^2{\theta} = 1$$
Confunction Identities
$$\sin{\left(\frac{\pi}{2} - \theta \right)} = \cos{\theta}$$ $$\tan{\left(\frac{\pi}{2} - \theta \right)} = \cot{\theta}$$ $$\csc{\left(\frac{\pi}{2} - \theta \right)} = \sec{\theta}$$ $$\cos{\left(\frac{\pi}{2} - \theta \right)} = \sin{\theta}$$ $$\cot{\left(\frac{\pi}{2} - \theta \right)} = \tan{\theta}$$ $$\sec{\left(\frac{\pi}{2} - \theta \right)} = \csc{\theta}$$
Double Angle Identities
$$\sin{2 \theta} = 2 \sin{\theta} \cos{\theta}$$ $$\cos{2 \theta} = \cos^2{\theta} - \sin^2{\theta}$$ $$\cos{2 \theta} = 2 \cos^2{\theta} - 1$$ $$\cos{2 \theta} = 1 - 2 \sin^2{\theta}$$ $$\tan{2 \theta} = \frac{2 \tan{\theta}}{1 - \tan^2{\theta}}$$
Triple Angle Identities
$$\sin{3 \theta} = 3 \sin{\theta} - 4 \sin^3{\theta}$$ $$\cos{3 \theta} = 4 \cos^3{\theta} - 3 \cos{\theta}$$ $$\tan{3 \theta} = \frac{3 \tan{\theta} - \tan^3{\theta}}{1 - 3 \tan^2{\theta}}$$
Sum to Product of Two Angles
$$\sin{\theta} + \sin{\phi} = 2 \sin{\left( \frac{\theta + \phi}{2} \right)} \cos{\left( \frac{\theta - \phi}{2} \right)}$$ $$\sin{\theta} - \sin{\phi} = 2 \cos{\left( \frac{\theta + \phi}{2} \right)} \sin{\left( \frac{\theta - \phi}{2} \right)}$$ $$\cos{\theta} + \cos{\phi} = 2 \cos{\left( \frac{\theta + \phi}{2} \right)} \cos{\left( \frac{\theta - \phi}{2} \right)}$$ $$\cos{\theta} - \cos{\phi} = -2 \sin{\left( \frac{\theta + \phi}{2} \right)} \sin{\left( \frac{\theta - \phi}{2} \right)}$$
Reciprocal Identities
$$\sin{\theta} = \frac{1}{\csc{\theta}}$$ $$\cos{\theta} = \frac{1}{\sec{\theta}}$$ $$\tan{\theta} = \frac{1}{\cot{\theta}}$$ $$\csc{\theta} = \frac{1}{\sin{\theta}}$$ $$\sec{\theta} = \frac{1}{\cos{\theta}}$$ $$\cot{\theta} = \frac{1}{\tan{\theta}}$$
Even/Odd Identities
$$\sin{(-\theta)} = - \sin{\theta}$$ $$\tan{(-\theta)} = - \tan{\theta}$$ $$\csc{(-\theta)} = - \csc{\theta}$$ $$\cos{(-\theta)} = \cos{\theta}$$ $$\cot{(-\theta)} = - \cot{\theta}$$ $$\sec{(-\theta)} = \sec{\theta}$$
Sum/Difference Identities
$$\sin{(\phi \pm \theta)} = \sin{\theta} \cos{\phi} \pm \cos{\theta} \sin{\phi}$$ $$\cos{(\theta \pm \phi)} = \cos{\theta} \cos{\phi} \mp \sin{\theta} \sin{\phi}$$ $$\tan{(\theta \pm \phi)} = \frac{\tan{\theta} \pm \tan{\phi}}{1 \mp \tan{\theta} \tan{\phi}}$$
Half Angle Identities
$$\sin^2{\theta} = \frac{1-\cos{(2 \theta)}}{2}$$ $$\cos^2{\theta} = \frac{1 + \cos{(2 \theta)}}{2}$$ $$\tan^2{\theta} = \frac{1 - \cos{(2 \theta)}}{1 + \cos{(2 \theta)}}$$
Product to Sum of Two Angles
$$2\sin{\theta} \sin{\phi} = \cos{(\theta - \phi)} - \cos{(\theta + \phi)}$$ $$2\cos{\theta} \cos{\phi} = \cos{(\theta - \phi)} + \cos{(\theta + \phi)}$$ $$2\sin{\theta} \cos{\phi} = \sin{(\theta + \phi)} + \sin{(\theta - \phi)}$$ $$2\cos{\theta} \sin{\phi} = \sin{(\theta + \phi)} - \sin{(\theta - \phi)}$$