Trigonometric identities

$$\cases{{}\\ Cos\ -φ=Cos\ φ=Cos\left({π±φ}\right)\\ {}\\ Sin\ -φ=-Sin\ φ=Sin\left({π+φ}\right)\\ {}\\ \qquad \qquad \qquad Sin\ \mathbf{φ}=Sin\left({π-\mathbf{φ}}\right)\\ {}\\ Cos\ φ=Sin\left({φ+\frac π2}\right)\ ,\ Cos\frac π4=Sin\frac{3π}4=Sin\left({π-\mathbf{\frac π4}}\right)=Sin\mathbf{\frac π4}\\ {}\\ Sin\ Θ=Cos\left({Θ-\frac π2}\right)\ ,\ Sin\frac π4=Cos\left({-\frac π4}\right)=Cos\frac π4\\ {}}$$

$$\displaystyle{Sin\ x=\sum_{i=1}^n\left({-1}\right)^{n-1}\frac{x^{2n-1}}{\left({2n-1}\right)!}}$$

$$\displaystyle{Cos\ x=1-\sum_{i=1}^n\left({-1}\right)^{n-1}\frac{x^{2n}}{\left({2n}\right)!}}$$

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  $$Sin\left({α±ß}\right)=Sin\ α·Cos\ ß±Cos\ α·Sin\ ß$$

  $$Cos\left({α±ß}\right)=Cos\ α·Cos\ ß∓Sin\ α·Sin\ ß$$

  $$Tan\left({α±ß}\right)=\frac{Tan\ α±Tan\ ß}{1∓Tan\ α·Tan\ ß}$$

  $$Cot\left({α±ß}\right)=\frac{Cot\ \mathbf{ß}·Cot\ \mathbf{α}∓1}{Cot\ \mathbf{ß}±Cot\ \mathbf{α}}$$

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$$Sin\ α±Sin\ ß=2·Sin\frac{α±ß}2·Cos\frac{α∓ß}2$$

$$Cos\ α±Cos\ ß=2·{}^{Cos}_{Sin}\frac{α+ß}2·{}^{Cos}_{Sin}\frac{α-ß}2$$

$$Tan\ α±Tan\ ß=\frac{Sin\left({α±ß}\right)}{Cos\ α·Cos\ ß}$$

  $$Cot\ α±Cot\ ß=\frac{Sin\left({\mathbf{ß}±\mathbf{α}}\right)}{Sin\ \mathbf{ß}·Sin\ \mathbf{α}}$$

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$$Sin^2\ α+Cos^2\ α=1=\\ =\left({Cos\ α-i·Sin\ α}\right)\left({Cos\ α+i·Sin\ α}\right)=\\ =\left({ch\ iα-sh\ iα}\right)\left({ch\ iα+sh\ iα}\right)=\\ =ch^2\ φ-sh^2\ φ$$

$$Tan^k\ α=Cot^{-k}\ α=\frac{Sin^k\ α}{Cos^k\ α}$$

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$$Sin\left({2α}\right)=2·Sin\ α·Cos\ α$$

$$Cos\left({2α}\right)=Cos^2\ α-Sin^2\ α=\\ =2·Cos^2\ α-1=1-2·Sin^2\ α$$

$$Tan\left({2α}\right)=\frac{2·Tan\ α}{1-Tan^2\ α}$$

$$Cot\left({2α}\right)=\frac{Cot^2\ α-1}{2·Cot\ α}$$

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$$1-Cos\ α=2·Sin^2\frac α2$$

$$1+Cos\ α=2·Cos^2\frac α2$$

$$1-Sin\ α=\left({Cos\frac α2-Sin\frac α2}\right)^2$$

$$1+Sin\ α=\left({Cos\frac α2+Sin\frac α2}\right)^2$$

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$$Sin\frac α2=±\sqrt{\frac{1-Cos\ α}2}$$

$$Cos\frac α2=±\sqrt{\frac{1+Cos\ α}2}$$

$$Tan\frac α2=±\sqrt{\frac{1-Cos\ α}{1+Cos\ α}}=\frac{1-Cos\ α}{Sin\ α}=\frac{Sin\ α}{1+Cos\ α}$$

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$$1+Tan^2\ α=\frac1{Cos^2\ α}$$

$$1+Cot^2\ α=\frac1{Sin^2\ α}$$

$$Sec\ α=\frac1{Cos\ α}=±\sqrt{1+Tan^2\ α}$$

$$Cosec\ α=\frac1{Sin\ α}=±\sqrt{1+Cot^2\ α}$$

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.	$Sin\ α$	$Cos\ α$	$Tan\ α$	$Cot\ α$
$Sin\ α$	.	$\left({1-Cos^2\ α}\right)^{\frac12}$	$\left({Tan^{-2}\ α+1}\right)^{-\frac12}$	$\left({1+Cot^2\ α}\right)^{-\frac12}$.
$Cos\ α$	$\left({1-Sin^2\ α}\right)^{\frac12}$	.	$\left({1+Tan^2\ α}\right)^{-\frac12}$	$\left({1+Cot^{-2}\ α}\right)^{-\frac12}$
$Tan\ α$	$\left({Sin^{-2}\ α-1}\right)^{-\frac12}$	$\left({Cos^2\ α-1}\right)^{\frac12}$	.	$Cot^{-1}\ α$
$Cot\ α$	$\left({Sin^{-2}\ α-1}\right)^{\frac12}$	$\left({Cos^2\ α-1}\right)^{-\frac12}$	$Tan^{-1}\ α$	.

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$Cos\ φ$	$0$	$-1$	$0$	$1$	$Cos\ φ$	$\frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}2$	$\frac{\sqrt{2+\sqrt{3}}}2$	$\frac{\sqrt{2+\sqrt{2}}}2$	$\frac{\sqrt{3}}2$	$\frac{\sqrt{2+\sqrt{2-\sqrt{3}}}}2$	$\frac{\sqrt{2}}2$	$\frac{\sqrt{2-\sqrt{2-\sqrt{3}}}}2$	$\frac 12$	$\frac{\sqrt{2-\sqrt{2}}}2$	$\frac{\sqrt{2-\sqrt{3}}}2$	$\frac{\sqrt{2-\sqrt{2+\sqrt{3}}}}2$	$Cos\ φ$
$Sin\ φ$	$-1$	$0$	$1$	$0$	$Sin\ φ$	$\frac{\sqrt{2-\sqrt{2+\sqrt{3}}}}2$	$\frac{\sqrt{2-\sqrt{3}}}2$	$\frac{\sqrt{2-\sqrt{2}}}2$	$\frac 12$	$\frac{\sqrt{2-\sqrt{2-\sqrt{3}}}}2$	$\frac{\sqrt{2}}2$	$\frac{\sqrt{2+\sqrt{2-\sqrt{3}}}}2$	$\frac{\sqrt{3}}2$	$\frac{\sqrt{2+\sqrt{2}}}2$	$\frac{\sqrt{2+\sqrt{3}}}2$	$\frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}2$	$Sin\ φ$
$φ$	$\frac{3π}2$	$π$	$\frac π2$	$2kπ$	$φ$	$\frac{π}{24}$	$\frac{π}{12}$	$\frac π8$	$\frac π6$	$\frac{5π}{24}$	$\frac π4$	$\frac{7π}{24}$	$\frac π3$	$\frac{9π}{24}$	$\frac{5π}{12}$	$\frac{11π}{24}$	$φ$
$Tan\ φ$	$±∞$	$0$	$±∞$	$0$	$Tan\ φ$	$\frac{\sqrt{2-\sqrt{3}}}{2+\sqrt{2+\sqrt{3}}}$	$2-\sqrt{3}$	$\frac1{\sqrt{2}+1}$	$\frac1{\sqrt{3}}$	$\frac{\sqrt{2+\sqrt{3}}}{2+\sqrt{2-\sqrt{3}}}$	$1$	$\frac{\sqrt{2+\sqrt{3}}}{2-\sqrt{2-\sqrt{3}}}$	$\sqrt{3}$	$\frac1{\sqrt{2}-1}$	$2+\sqrt{3}$	$\frac{\sqrt{2-\sqrt{3}}}{2-\sqrt{2+\sqrt{3}}}$	$Tan\ φ$
$Cot\ φ$	$0$	$±∞$	$0$	$±∞$	$Cot\ φ$	$\frac{\sqrt{2-\sqrt{3}}}{2-\sqrt{2+\sqrt{3}}}$	$2+\sqrt{3}$	$\frac1{\sqrt{2}-1}$	$\sqrt{3}$	$\frac{\sqrt{2+\sqrt{3}}}{2-\sqrt{2-\sqrt{3}}}$	$1$	$\frac{\sqrt{2+\sqrt{3}}}{2+\sqrt{2-\sqrt{3}}}$	$\frac1{\sqrt{3}}$	$\frac1{\sqrt{2}+1}$	$2-\sqrt{3}$	$\frac{\sqrt{2-\sqrt{3}}}{2+\sqrt{2+\sqrt{3}}}$	$Cot\ φ$

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$$arcCos\ A=±i·ln\left({A±\sqrt{A^2-1}}\right)$$

$$arcSin\ A=\left({±}\right)\ i·ln\left({±\sqrt{1-A^2}+\left({∓}\right)\ i·A}\right)$$

$$\frac i2\ ln\ \frac{1-i·A}{1+i·A}=arcTan\ A=\frac i2\ ln\ \frac{i+A}{i-A}$$

  $$\frac i2\ ln\ \frac{i·A+1}{i·A-1}=arcCot\ A=\frac i2\ ln\ \frac{A-i}{A+i}$$

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$$arcTan\ A+arcCot\ A=\frac π2±2kπ$$

$$±\left({arcTan\ A-arcCot\ A}\right)=\frac i2\ ln\ \left({i·\frac{i·A∓1}{i·A±1}}\right)^2=\frac i2\ ln\ \left({-i·\frac{1∓i·A}{1±i·A}}\right)^2=\frac i2\ ln\ \left({i·\frac{A±i}{A∓i}}\right)^2=etc. ...$$

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