Trigonometric identities or relationships between trigonometric functions are as follows:
\begin{align} \sin^{2}\alpha+\cos^{2}\alpha &=(\sin \alpha)^{2}+(\cos \alpha)^{2}=1\\ \\ 1+\tan^{2}\alpha &=\sec^{2}\alpha=\frac{1}{\cos^{2}\alpha} \\ \\ 1+\cot^{2}\alpha &=\csc^{2}\alpha=\frac{1}{\sin^{2}\alpha} \\ \end{align}
The above are also called Pythagorean identities because they can be viewed as a special case of the Pythagorean theorem, where:
$$a^{2}+b^{2}=1$$
This can be applied, for example, to the unit circle where the radius of the circle is r=1.
From the cotangent function, you can also derive the identity:
\begin{align} \cot \alpha &=\frac{1}{\tan \alpha} \\ \\ &\Rightarrow \\ \\ \tan \alpha \times \cot \alpha&=1\\ \end{align}