Summary of Trigonometric Identities

Basic Identities
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1. $\sin \theta = \dfrac{1}{\csc \theta}; \,\, \csc \theta = \dfrac{1}{\sin \theta}$
2. $\cos \theta = \dfrac{1}{\sec \theta}; \,\, \sec \theta = \dfrac{1}{\cos \theta}$
3. $\tan \theta = \dfrac{\sin \theta}{\cos \theta} = \dfrac{1}{\cot \theta}$
4. $\cot \theta = \dfrac{\cos \theta}{\sin \theta} = \dfrac{1}{\tan \theta}$
 

Pythagorean Identities
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1. $\sin^2 \theta + \cos^2 \theta = 1$
2. $\tan^2 \theta + 1 = \sec^2 \theta$
3. $1 + \cot^2 \theta = \csc^2 \theta$
 

Sum and Difference of Two Angles
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1. $\sin (\alpha + \beta) = \sin \alpha \, \cos \beta + \cos \alpha \, \sin \beta$
2. $\sin (\alpha - \beta) = \sin \alpha \, \cos \beta - \cos \alpha \, \sin \beta$
3. $\cos (\alpha + \beta) = \cos \alpha \, \cos \beta - \sin \alpha \, \sin \beta$
4. $\cos (\alpha - \beta) = \cos \alpha \, \cos \beta + \sin \alpha \, \sin \beta$
5. $\tan (\alpha + \beta) = \dfrac{\tan \alpha + \tan \beta}{1 - \tan \alpha \, \tan \beta}$
6. $\tan (\alpha - \beta) = \dfrac{\tan \alpha - \tan \beta}{1 + \tan \alpha \, \tan \beta}$
 

Double Angle Formulas
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1. $\sin 2\theta = 2 \sin \theta \, \cos \theta$
2. $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 1 - 2\sin^2 \theta = 2\cos^2 \theta - 1$
3. $\tan 2\theta = \dfrac{2\tan \theta}{1 - \tan^2 \theta}$
 

Half Angle Formulas
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1. $\sin \frac{1}{2}\theta = \sqrt{\dfrac{1 - \cos \theta}{2}}$
2. $\cos \frac{1}{2}\theta = \sqrt{\dfrac{1 + \cos \theta}{2}}$
3. $\tan \frac{1}{2}\theta = \dfrac{1 - \cos \theta}{\sin \theta} = \dfrac{\sin \theta}{1 + \cos \theta} = \sqrt{\dfrac{1 - \cos \theta}{1 + \cos \theta}}$
 

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