# Basic Trigonometric Identities

• Angle: $$\alpha$$
Trigonometric functions: $$\sin \alpha,$$ $$\cos \alpha,$$ $$\tan \alpha,$$ $$\cot \alpha,$$ $$\sec \alpha,$$ $$\csc \alpha$$
Set of integers: $$\mathbb{Z}$$
Integers: $$n$$
1. Trigonometric identities establish a connection between trigonometric functions of the same argument (angle $$\alpha$$).
2. Pythagorean trigonometric identity
$${\sin ^2}\alpha + {\cos ^2}\alpha = 1$$
This identity is the result of application of the Pythagorean theorem to a triangle in the unit circle.
3. Relationship between the cosine and tangent
$$\large{\frac{1}{{\cos^2}\alpha}}\normalsize – {\tan ^2}\alpha = 1$$ or $$\sec^2\alpha – {\tan ^2}\alpha = 1.$$
This identity follows from the Pythagorean trigonometric identity and is obtained by dividing the left and right sides by $$\cos^2 \alpha$$. It is assumed that $$\alpha \ne \large{\frac{\pi}{2}}\normalsize + \pi n,$$ $$n \in \mathbb{Z}$$.
4. Relationship between the sine and cotangent
$$\large{\frac{1}{{\sin^2}\alpha}}\normalsize – {\cot ^2}\alpha = 1$$ or $$\csc^2\alpha – {\cot ^2}\alpha = 1.$$
This formula also follows from the Pythagorean trigonometric identity (it is obtained by dividing the left and right sides by $$\sin^2 \alpha$$. It is assumed that $$\alpha \ne \pi n,$$ $$n \in \mathbb{Z}$$.
5. Definition of tangent
$$\tan \alpha = \large{\frac{\sin \alpha}{\cos \alpha}}\normalsize ,$$ where $$\alpha \ne \large{\frac{\pi}{2}}\normalsize + \pi n,$$ $$n \in \mathbb{Z}$$.
6. Definition of cotangent
$$\cot \alpha = \large{\frac{\cos \alpha}{\sin \alpha}}\normalsize ,$$ where $$\alpha \ne \pi n,$$ $$n \in \mathbb{Z}$$.
7. Consequence of the definitions of tangent and cotangent
$$\tan \alpha \cdot \cot \alpha = 1,$$ where $$\alpha \ne \large{\frac{\pi n}{2}}\normalsize ,$$ $$n \in \mathbb{Z}.$$
8. Definition of secant
$$\sec \alpha = \large{\frac{1}{\cos \alpha}}\normalsize,$$ $$\alpha \ne \large{\frac{\pi}{2}}\normalsize +\pi n,$$ $$n \in \mathbb{Z}.$$
9. Definition of cosecant
$$\csc \alpha = \large{\frac{1}{\sin \alpha}}\normalsize,$$ $$\alpha \ne \pi n,$$ $$n \in \mathbb{Z}.$$