Trig calculus

The derivations for these identities are quite insane, and must be studied at a later date, when sanity is abundant and assignments are few.

Differentiation

$\sin{x}\Rightarrow\cos{x}$ $\cos{x}\Rightarrow-\sin{x}$ $\tan{x}\Rightarrow\sec^{2}{x}$ $\sec{x}\Rightarrow\sec{x}\tan{x}$ $\csc{x}\Rightarrow-\csc{x}\cot{x}$ $\cot{x}\Rightarrow-\csc^{2}{x}$ $\arctan{x}\Rightarrow\frac{1}{1+x^{2}}$

$\arcsin{x}\Rightarrow\frac{1}{\sqrt{1-x^{2}}}$ $\arccos{x}\Rightarrow-\frac{1}{1-x^{2}}$ $\sinh{x}\Rightarrow\cosh{x}$ $\cosh{x}\Rightarrow\sinh{x}$ $\tanh{x}\Rightarrow\sech^{2}{x}$ $\arcsinh{x}\Rightarrow\frac{1}{\sqrt{1+x^{2}}}$ $\arccosh{x}\Rightarrow\frac{1}{\sqrt{x^{2}-1}}$ $\arctanh{x}\Rightarrow\frac{1}{1-x^{2}}$

Integration

$\int{\frac{1}{x^{2}+a^{2}}}=\frac{1}{a}\arctan{x}$