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⇒cos⁡x\sin{x}\Rightarrow\cos{x} cos⁡x⇒−sin⁡x\cos{x}\Rightarrow-\sin{x} tan⁡x⇒sec⁡2x\tan{x}\Rightarrow\sec^{2}{x} sec⁡x⇒sec⁡xtan⁡x\sec{x}\Rightarrow\sec{x}\tan{x} csc⁡x⇒−csc⁡xcot⁡x\csc{x}\Rightarrow-\csc{x}\cot{x} cot⁡x⇒−csc⁡2x\cot{x}\Rightarrow-\csc^{2}{x} arctan⁡x⇒11+x2\arctan{x}\Rightarrow\frac{1}{1+x^{2}} arcsin⁡x⇒11−x2\arcsin{x}\Rightarrow\frac{1}{\sqrt{1-x^{2}}} arccos⁡x⇒−11−x2\arccos{x}\Rightarrow-\frac{1}{1-x^{2}} sinh⁡x⇒cosh⁡x\sinh{x}\Rightarrow\cosh{x} cosh⁡x⇒sinh⁡x\cosh{x}\Rightarrow\sinh{x} tanh⁡x⇒sech⁡2x\tanh{x}\Rightarrow\sech^{2}{x} arcsinh⁡x⇒11+x2\arcsinh{x}\Rightarrow\frac{1}{\sqrt{1+x^{2}}} arccosh⁡x⇒1x2−1\arccosh{x}\Rightarrow\frac{1}{\sqrt{x^{2}-1}} arctanh⁡x⇒11−x2\arctanh{x}\Rightarrow\frac{1}{1-x^{2}}

Integration

∫1x2+a2=1aarctan⁡x\int{\frac{1}{x^{2}+a^{2}}}=\frac{1}{a}\arctan{x}