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}
See also:
Trigonometric Identities FAQ
What are trigonometric identities?
They are equations that are true for every valid angle value.
Why are Pythagorean identities important?
They connect sine, cosine, tangent, secant, and cotangent through fundamental relationships.
