$! sin^2\alpha+cos^2\alpha=1; $
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$! 1+ctg^2\alpha=\displaystyle\frac{1}{sin^2\alpha};$
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$! 1+tg^2\alpha=\displaystyle\frac{1}{cos^2\alpha}\cdot$
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$! {sin({\alpha}{\ \pm\ }{\beta})}=sin\alpha{\ \cdot\ }cos\beta{\ \pm\ }cos\alpha{\ \cdot\ }sin\beta; $
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$! {cos({\alpha}{\ \pm\ }{\beta})}=cos\alpha{\ \cdot\ }cos\beta{\ \mp\ }sin\alpha{\ \cdot\ }sin\beta; $
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$! tg({\alpha\ \pm\ }\beta)= \displaystyle\frac{tg\alpha{\ \pm\ }tg\beta}{1{\ \mp\ }tg\alpha{\ \cdot\ }tg\beta};$

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$! ctg({\alpha\ \pm\ }\beta)= \displaystyle\frac{ctg\alpha{\ \cdot\ }ctg\beta {\ \mp\ }1}{ctg\beta{\ \pm\ }ctg\alpha} {\cdot} $
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$! sin2\alpha=2sin\alpha{\ \cdot\ }cos\alpha; $
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$! cos2\alpha =cos^2\alpha-sin^2\alpha; $
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$! tg2\alpha =\displaystyle\frac{2\ tg\alpha}{1-tg^2\alpha}{\cdot} $
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$! sin\alpha{\ \cdot\ }sin\beta={\displaystyle\frac{1}{2}} \bigl(cos(\alpha-\beta)-cos(\alpha+\beta)\bigr);$
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$! cos\alpha{\ \cdot\ }cos\beta={\displaystyle\frac{1}{2}} \bigl(cos(\alpha+\beta)+cos(\alpha-\beta)\bigr);$
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$! sin\alpha{\ \cdot\ }cos\beta={\displaystyle\frac{1}{2}} \bigl(sin(\alpha+\beta)+sin(\alpha-\beta)\bigr).$
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$! sin\alpha+sin\beta=2\ sin{\displaystyle\frac{\alpha+\beta}{2}}{\ \cdot\ }cos{\displaystyle\frac{\alpha-\beta}{2}};\hspace{18pt}
cos\alpha+cos\beta=2\ cos{\displaystyle\frac{\alpha+\beta}{2}}{\ \cdot\ }cos{\displaystyle\frac{\alpha-\beta}{2}};$
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$! sin\alpha-sin\beta=2\ cos{\displaystyle\frac{\alpha+\beta}{2}}{\ \cdot\ }sin{\displaystyle\frac{\alpha-\beta}{2}};\hspace{18pt}
cos\alpha-cos\beta=-2\ sin{\displaystyle\frac{\alpha+\beta}{2}}{\ \cdot\ }sin{\displaystyle\frac{\alpha-\beta}{2}}\cdot$
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$! sin x=a,\hspace{140pt} x=(-1)^k\ arcsin{\ a}+{\pi}k; $
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$! cos x=a,\hspace{140pt} x={\pm}arccos{\ }a+2{\pi}k; $
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$! tg x=a,\hspace{145pt} x=arctg{\ }a+{\pi}k. $