These identities define key relationships between trigonometric functions and are essential in simplifying, solving, and transforming trigonometric equations.
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]
Explanation: This is the fundamental Pythagorean identity. It arises from the Pythagorean theorem applied to a unit circle, where the radius (hypotenuse) is 1.
\[ \tan \alpha \cdot \cot \alpha = 1 \]
Explanation: Tangent and cotangent are reciprocals of each other. So, multiplying them always results in 1: \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\), \(\cot \alpha = \frac{\cos \alpha}{\sin \alpha}\).
\[ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{1}{\cot \alpha} \]
Explanation: This is a quotient identity. Tangent is the ratio of sine to cosine, and it is the reciprocal of cotangent.
\[ \cot \alpha = \frac{\cos \alpha}{\sin \alpha} = \frac{1}{\tan \alpha} \]
Explanation: Cotangent is the reciprocal of tangent. It's defined as the ratio of cosine to sine.
\[ 1 + \tan^2 \alpha = \frac{1}{\cos^2 \alpha} = \sec^2 \alpha \]
Explanation: This identity comes from dividing the Pythagorean identity by \(\cos^2 \alpha\). It links tangent and secant functions.
\[ 1 + \cot^2 \alpha = \frac{1}{\sin^2 \alpha} = \csc^2 \alpha \]
Explanation: This identity is derived by dividing the Pythagorean identity by \(\sin^2 \alpha\), relating cotangent and cosecant.