$$\cos^2\theta+\sin^2\theta=1$$ $$1+\tan^2\theta=\sec^2\theta$$ $$1+\cot^2\theta=\csc^2\theta$$ $$\sin^2\theta=1-\cos^2\theta=\frac{1-\cos2\theta}{2}$$ $$\cos^2\theta=1-\sin^2\theta=\frac{1+\cos2\theta}{2}$$

$$\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$$ $$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$$ $$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$$

$$\cos2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A$$ $$\sin2A=2\sin A\cos A$$ $$\tan2A=\frac{2\tan A}{1-\tan^2A}$$

$$\cos\tfrac{A}{2}=\pm\sqrt{\frac{1+\cos A}{2}}$$ $$\sin\tfrac{A}{2}=\pm\sqrt{\frac{1-\cos A}{2}}$$ $$\tan\tfrac{A}{2}=\pm\sqrt{\frac{1-\cos A}{1+\cos A}}=\frac{1-\cos A}{\sin A}=\frac{\sin A}{1+\cos A}$$

$$\cos(\tfrac{\pi}{2}-A)=\sin A$$ $$\sin(\tfrac{\pi}{2}-A)=\cos A$$ $$\tan(\tfrac{\pi}{2}-A)=\cot A$$ $$\cot(\tfrac{\pi}{2}-A)=\tan A$$ $$\sec(\tfrac{\pi}{2}-A)=\csc A$$ $$\csc(\tfrac{\pi}{2}-A)=\sec A$$

$$\cos(-\theta)=\cos(\theta)$$ $$\sin(-\theta)=-1\cdot\sin(\theta)$$ $$\tan(-\theta)=-1\cdot\tan(\theta)$$ $$\cot(-\theta)=-1\cdot\cot(\theta)$$ $$\csc(-\theta)=-1\cdot\csc(\theta)$$ $$\sec(-\theta)=\sec(\theta)$$

$$\frac{\mathrm{d} }{\mathrm{d} x}\sin x=\cos x$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\cos x=-\sin x$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\tan x=\sec^2x$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\csc x=-\csc x\cot x$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\sec x=\sec x\tan x$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\cot x=-\csc^2x$$

$$\frac{\mathrm{d} }{\mathrm{d} x}\sin^{-1}x=\frac{1}{\sqrt{1-x^2}}$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\tan^{-1}x=\frac{1}{{1+x^2}}$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\cot^{-1}x=-\frac{1}{{1+x^2}}$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\sec^{-1}x=\frac{1}{x\sqrt{x^2-1}}$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\csc^{-1}x=-\frac{1}{x\sqrt{x^2-1}}$$