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##### Trigonometric Identities

Sine, tangent, cotangent and cosecant in mathematics an identity is an equation that is always true. Meanwhile trigonometric identities are equations that involve trigonometric functions that are always true.

This identities mostly refer to one angle labelled $\displaystyle \theta$.

### Defining Tangent, Cotangent, Secant and Cosecant from Sine and Cosine

$\displaystyle \tan \theta =\frac{{\sin \theta }}{{\cos \theta }}$

$\displaystyle \cot \theta =\frac{1}{{\tan \theta }}=\frac{{\cos \theta }}{{\sin \theta }}$

$\displaystyle \sec \theta =\frac{1}{{\cos \theta }}$

$\displaystyle \csc \theta =\frac{1}{{\sin \theta }}$

### Trigonometry Table

The table with the most used angles in trigonometry.

### Pythagorean Identities

The most important trigonometric identity

$\displaystyle {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$

Less used identities

$\displaystyle {{\sec }^{2}}\theta =1+{{\tan }^{2}}\theta$

$\displaystyle {{\csc }^{2}}\theta =1+{{\cot }^{2}}\theta$

### Complement Angles Identities

$\displaystyle \sin \theta =\cos \left( {{{{90}}^{\circ }}-\theta } \right)$

$\displaystyle \cos \theta =\sin \left( {{{{90}}^{\circ }}-\theta } \right)$

$\displaystyle \tan \theta =\cot \left( {{{{90}}^{\circ }}-\theta } \right)$

$\displaystyle \cot \theta =\tan \left( {{{{90}}^{\circ }}-\theta } \right)$

$\displaystyle \sec \theta =\csc \left( {{{{90}}^{\circ }}-\theta } \right)$

$\displaystyle \csc \theta =\sec \left( {{{{90}}^{\circ }}-\theta } \right)$

Note! You can find the angle written with $\displaystyle \pi$, reciprocally $\displaystyle {{90}^{\circ }}=\frac{\pi }{2}$.

### Supplement Angles Identities

$\displaystyle \sin \left( {{{{180}}^{\circ }}-\theta } \right)=\sin \theta$

$\displaystyle \cos \left( {{{{180}}^{\circ }}-\theta } \right)=-\cos \theta$

$\displaystyle \tan \left( {{{{180}}^{\circ }}-\theta } \right)=-\tan \theta$

$\displaystyle \cot \left( {{{{180}}^{\circ }}-\theta } \right)=-\cot \theta$

$\displaystyle \sec \left( {{{{180}}^{\circ }}-\theta } \right)=-\sec \theta$

$\displaystyle \csc \left( {{{{180}}^{\circ }}-\theta } \right)=\csc \theta$

Note! You can find the angle written with $\displaystyle \pi$, reciprocally $\displaystyle {{180}^{\circ }}=\pi$

### Periodicity Identities

Sine, cosine, secant and cosecant have periods $\displaystyle {{360}^{\circ }}(2\pi )$ while tangent and  cotangent have periods $\displaystyle {{180}^{\circ }}(\pi )$.

$\displaystyle \sin (\theta +{{360}^{\circ }})=\sin \theta$

$\displaystyle \cos (\theta +{{360}^{\circ }})=\cos \theta$

$\displaystyle \sec (\theta +{{360}^{\circ }})=\sec \theta$

$\displaystyle \csc (\theta +{{360}^{\circ }})=\csc \theta$

$\displaystyle \tan (\theta +{{180}^{\circ }})=\tan \theta$

$\displaystyle \cot (\theta +{{180}^{\circ }})=\cot \theta$

### Negative Angles Identities

Sine, tangent, cotangent and cosecant are odd functions, while cosine and secant are even functions.

$\displaystyle \sin (-\theta )=-\sin \theta$

$\displaystyle \cos \theta =\sin \left( {{{{90}}^{\circ }}-\theta } \right)$

$\displaystyle \tan (-\theta )=-\tan \theta$

$\displaystyle \cot (-\theta )=-\cot \theta$

$\displaystyle \cos (-\theta )=\cos \theta$

$\displaystyle \sec (-\theta )=\sec \theta$

### Ptolemy’s Identities

The sum and difference of two angles identities

$\displaystyle \sin (A+B)=$$\displaystyle \sin A\cos B+\cos A\sin B \displaystyle \cos (A+B)=$$\displaystyle \cos A\cos B-\sin A\sin B$

$\displaystyle \tan (A+B)=$$\displaystyle \frac{{\tan A+\tan B}}{{1-\tan A\tan B}} \displaystyle \sin (A-B)=$$\displaystyle \sin A\cos B-\cos A\sin B$

$\displaystyle \cos (A-B)=$$\displaystyle \cos A\cos B+\sin A\sin B \displaystyle \tan (A-B)=$$\displaystyle \frac{{\tan A-\tan B}}{{1+\tan A\operatorname{Tan}B}}$

### Sum/Difference into Product Identities

These identities allow us to change a sum or difference of sines or cosinesinto a product of sines or cosines.

$\displaystyle \sin A+\sin B=$$\displaystyle 2\sin \frac{{A+B}}{2}\cos \frac{{A-B}}{2} \displaystyle \cos A+\cos B=$$\displaystyle 2\cos \frac{{A+B}}{2}\cos \frac{{A-B}}{2}$

$\displaystyle \sin A-\sin B=$$\displaystyle 2\cos \frac{{A+B}}{2}\sin \frac{{A+B}}{2} \displaystyle \cos A-\cos B=$$\displaystyle -2\sin \frac{{A+B}}{2}\sin \frac{{A-B}}{2}$

### Product into Sum/Difference Identities

$\displaystyle \sin A\cdot \sin B=$$\displaystyle \frac{{\cos (A-B)-\cos (A+B)}}{2} \displaystyle \cos A\cdot \cos B=$$\displaystyle \frac{{\cos (A-B)+\cos (A+B)}}{2}$

$\displaystyle \sin A\cdot \cos B=$$\displaystyle \frac{{\sin (A+B)+\sin (A-B)}}{2} \displaystyle \cos A\cdot \sin B=$$\displaystyle \frac{{\sin (A+B)-\sin (A-B)}}{2}$

### Double Angles Identities

$\displaystyle \sin 2\theta =2\sin \theta \cos \theta$

$\displaystyle \cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta$

$\displaystyle \tan 2\theta =\frac{{2\tan \theta }}{{1-{{{\tan }}^{2}}\theta }}$

### Triple Angle Identities

$\displaystyle \sin 3\theta =3\sin \theta -4{{\sin }^{3}}\theta$

$\displaystyle \cos 3\theta =4{{\cos }^{3}}\theta -3\cos \theta$

$\displaystyle \tan 3\theta =\frac{{3\tan \theta -{{{\tan }}^{3}}\theta }}{{1-3{{{\tan }}^{2}}\theta }}$

### Half Angle Identities

The first and second identities take minus or plus sign depending on the quadrant in which is the angle.

$\displaystyle \sin \frac{\theta }{2}=\pm \sqrt{{\frac{{1-\cos \theta }}{2}}}$

$\displaystyle \cos \frac{\theta }{2}=\pm \sqrt{{\frac{{1+\cos \theta }}{2}}}$

$\displaystyle \tan \frac{\theta }{2}=\frac{{\sin \theta }}{{1+\cos \theta }}=\frac{{1-\cos \theta }}{{\sin \theta }}$

### Power reducing or half angle Identities

$\displaystyle {{\sin }^{2}}\theta =\frac{{1-\cos 2\theta }}{2}$

$\displaystyle \sin \theta =\sqrt{{\frac{{1-\cos 2\theta }}{2}}}$

$\displaystyle {{\cos }^{2}}\theta =\frac{{1+\cos 2\theta }}{2}$

$\displaystyle \cos \theta =\sqrt{{\frac{{1+\cos 2\theta }}{2}}}$

$\displaystyle {{\tan }^{2}}\theta =\frac{{1-\cos 2\theta }}{{1+\cos 2\theta }}$

$\displaystyle \tan \theta =\sqrt{{\frac{{1-\cos 2\theta }}{{1+\cos 2\theta }}}}$

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