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Trigonometry - Relations between trigonometric functions

  1. \(sin\alpha=\pm\sqrt{1-cos^2\alpha}=\pm\sqrt{{1\over2}(1-cos2\alpha)}\) =\(2cos^2({\alpha \over 2}-{\pi \over 2})-1\)=\({2tan({\alpha \over 2})} \over 1+tan^2 {\alpha \over2}\)
  2. \(cos\alpha=\pm\sqrt{1-sin^2\alpha}=\pm\sqrt{{1\over2}(1+cos2\alpha)}\)\(2cos^2{\alpha \over 2}-1\)\({1-tan^2{\alpha \over 2}} \over 1+tan^2 {\alpha \over2}\)
  3. \(tan\alpha={sin\alpha \over cos\alpha}=\pm \sqrt{1-sin^2\alpha}\)=\(sin2\alpha \over1+cos2\alpha\)=\(1-cos2\alpha \over sin2\alpha\)=\(2tan{\alpha \over 2} \over1+ 2tan^2{\alpha \over 2}\)=\({2tan{\alpha \over2}} \over1+tan^2{\alpha \over 2}\)=\({\pm\sqrt{1-cos2\alpha \over 1+cos2\alpha}}\)
  4. \(cot\alpha={cos\alpha \over sin\alpha}\)=\(\pm\sqrt{sec^2-1}\)=\(1+cos2\alpha \over sin2 \alpha\)=\(\pm{\sqrt{{1+cos2\alpha} \over {1-cos2\alpha}}}\)=\({1-tan^2{\alpha \over2}} \over2tan^2{\alpha \over 2}\)
  5. \(sec \alpha= {1 \over cos \alpha}\)=\(\pm\sqrt{1+tan^2\alpha}\)=\({1+tan^2{\alpha \over2}} \over {1-tan^2{\alpha \over2}}\)
  6. \(cosec \alpha= {1 \over sin \alpha}\)=\(\pm \sqrt {1+cot^2 \alpha }\)=\({1+tan^2{\alpha \over2}} \over2tan{\alpha \over 2}\)