Two basic functions that you must memorize are:
$$$f(x) = \sin x \Rightarrow f'(x)=\cos x$$$ $$$f(x) = \cos x \Rightarrow f'(x)=-\sin x$$$
In addition to the two most basic functions, other trigonometric functions and its derivatives are:
$$$f (x) =\tan x \Rightarrow f'(x)= \frac{1}{\cos^2 x} $$$ $$$f (x) =\sec x \Rightarrow f'(x)= \sec x \cdot \tan x$$$ $$$f (x) =\csc x \Rightarrow f'(x)=-\csc x \cdot \cot x $$$ $$$f (x) =\cot x \Rightarrow f'(x)=- \csc^2 x $$$ $$$f (x) =\arcsin x \Rightarrow f'(x)=\frac{1}{\sqrt{1-x^2}}$$$ $$$f (x) =\arccos x \Rightarrow f'(x)=\frac{-1}{\sqrt{1-x^2}}$$$ $$$f (x) =\arctan x \Rightarrow f'(x)=\frac{1}{1+x^2}$$$
Some of the functions exhibited in this second block can be deduced from some rules of derivation. For example, being the tangent one the quotient between bosom and cosine, we might calculate its derivative from the derivatives already known about the bosom and the cosine using the rule of the quotient.