Trigonometric identities
Pythagorean
- \( \sin^2 \theta + \cos^2 \theta = 1 \)
- \( \tan^2 \theta + 1 = \sec^2 \theta \)
- \( 1 + \cot^2 \theta = \csc^2 \theta \)
Double angle
- \( \sin^2 \theta = \frac{1}{2} - \frac{1}{2} \cos(2\theta) \)
- \( \cos^2 \theta = \frac{1}{2} + \frac{1}{2} \cos(2\theta) \)
- \( \sin 2\theta = 2 \sin \theta \cdot \cos \theta \)
- \( \cos 2\theta = 2 \cos^2 \theta - \sin^2 \theta \)
Product-sum
- \( \sin(A) \cos(B) = \frac{1}{2} \sin(A-B) + \frac{1}{2} \sin(A+B) \)
Derivatives
- \( \frac{d}{dx} \left( \csc \theta \right) = - \cot \theta \cdot \csc \theta \)
- \( \frac{d}{dx} \left( \sec \theta \right) = \tan \theta \cdot \sec \theta \)
- \( \frac{d}{dx} \left( \cot \theta \right) = -\csc^2 \theta \)
Integrals
- \( \displaystyle\int \csc \theta \, d\theta = -\ln| \cot \theta + \csc \theta | + C \)
- \( \displaystyle\int \sec \theta \, d\theta = \ln| \tan \theta + \sec \theta | + C \)
- \( \displaystyle\int \cot \theta \, d\theta = \ln|\sin \theta| + C \)