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Fundamental Identities

Quotient and reciprocal identities
  • $ \tan(x)=\frac{\sin(x)}{\cos(x)} $
  • $ \cot(x)=\frac{\cos(x)}{\sin(x)}=\frac{\csc(x)}{\sec(x)}=\frac{1}{\tan(x)} $
  • $ \sec\theta=\frac{1}{\cos(x)} $
  • $ \csc\theta=\frac{1}{\sin(x)} $
Pythagorean identities
  • $ \sin^2(x)+\cos^2(x)=1 $
  • $ \tan^2(x)+1=\sec^2(x) $
  • $ 1+ \cot^2(x)=\csc^2(x) $
Angle sum and difference identities
  • $ \sin(x\pm y)=\sin(x)\cos(y)\pm\sin(y)\cos(x) $
  • $ \cos(x\pm y)=\cos(x)\cos(y)\mp\sin(x)\sin(y) $
  • $ \tan(x\pm y)=\frac{\tan(x)\pm\tan(y)}{1\mp\tan(x)\tan(y)} $
Double-angle identities
  • $ \sin(2x)=2\sin(x)\cos(x) $
  • $ \cos(2x)=\cos^2(x)-\sin^2(x)=2\cos^2(x)-1=1-2\sin^2(x) $
  • $ \tan(2x)=\frac{2\tan(x)}{1-\tan^2(x)} $
Half-angle identities
  • $ \sin\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1-\cos(x)}{2}} $
  • $ \cos\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1+\cos(x)}{2}} $
  • $ \tan\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}=\frac{\sin(x)}{1+\cos(x)}=\frac{1-\cos(x)}{\sin(x)} $
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