2
tg
α
1 +
tg
2
α
,
{\displaystyle \sin 2\alpha =2\sin \alpha \cos \alpha ={\frac {2,\operatorname {tg} ,\alpha }{1+\operatorname {tg} ^{2}\alpha }},}
cos
2
α
−
sin
2
2
cos
2
α
−
1
−
2
sin
2
1 −
tg
2
α
1 +
tg
2
α
=
ctg
α − tg
α
ctg
α + tg
α
,
{\displaystyle \cos 2\alpha =\cos ^{2}\alpha ,-,\sin ^{2}\alpha =2\cos ^{2}\alpha ,-,1=1,-,2\sin ^{2}\alpha ={\frac {1-\operatorname {tg} ^{2}\alpha }{1+\operatorname {tg} ^{2}\alpha }}={\frac {\operatorname {ctg} ,\alpha -\operatorname {tg} ,\alpha }{\operatorname {ctg} ,\alpha +\operatorname {tg} ,\alpha }},}
tg
2
tg
α
1 −
tg
2
α
,
{\displaystyle \operatorname {tg} ,2\alpha ={\frac {2,\operatorname {tg} ,\alpha }{1-\operatorname {tg} ^{2}\alpha }},}
ctg
ctg
2
α − 1
2
ctg
α
=
1 2
(
ctg
α − tg
α
)
.
{\displaystyle \operatorname {ctg} ,2\alpha ={\frac {\operatorname {ctg} ^{2}\alpha -1}{2,\operatorname {ctg} ,\alpha }}={\frac {1}{2}}\left(\operatorname {ctg} ,\alpha -\operatorname {tg} ,\alpha \right).}
2 sin
α ± β
2
cos
α ∓ β
2
,
{\displaystyle \sin \alpha \pm \sin \beta =2\sin {\frac {\alpha \pm \beta }{2}}\cos {\frac {\alpha \mp \beta }{2}},}
2 cos
α + β
2
cos
α − β
2
,
{\displaystyle \cos \alpha +\cos \beta =2\cos {\frac {\alpha +\beta }{2}}\cos {\frac {\alpha -\beta }{2}},}
− 2 sin
α + β
2
sin
α − β
2
,
{\displaystyle \cos \alpha -\cos \beta =-2\sin {\frac {\alpha +\beta }{2}}\sin {\frac {\alpha -\beta }{2}},}
sin ( α ± β )
cos α cos β
,
{\displaystyle \operatorname {tg} \alpha \pm \operatorname {tg} \beta ={\frac {\sin(\alpha \pm \beta )}{\cos \alpha \cos \beta }},}
1 ± sin
2 α
= ( sin α ± cos α
)
2
.
{\displaystyle 1\pm \sin {2\alpha }=(\sin \alpha \pm \cos \alpha )^{2}.}