edit
cos
(
x
±
y
)
=
cos
(
x
)
cos
(
y
)
∓
sin
(
x
)
sin
(
y
)
{\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)}
sin
(
x
±
y
)
=
sin
(
x
)
cos
(
y
)
±
sin
(
y
)
cos
(
x
)
{\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \sin(y)\cos(x)}
tan
(
x
±
y
)
=
tan
(
x
)
±
tan
(
y
)
1
∓
tan
(
x
)
tan
(
y
)
{\displaystyle \tan(x\pm y)={\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}}
2
sin
(
x
)
sin
(
y
)
=
cos
(
x
−
y
)
−
cos
(
x
+
y
)
{\displaystyle 2\sin(x)\sin(y)=\cos(x-y)-\cos(x+y)}
2
cos
(
x
)
cos
(
y
)
=
cos
(
x
−
y
)
+
cos
(
x
+
y
)
{\displaystyle 2\cos(x)\cos(y)=\cos(x-y)+\cos(x+y)}
2
sin
(
x
)
cos
(
y
)
=
sin
(
x
−
y
)
+
sin
(
x
+
y
)
{\displaystyle 2\sin(x)\cos(y)=\sin(x-y)+\sin(x+y)}
Sum and difference to product
edit
A
sin
(
x
)
+
B
cos
(
x
)
=
C
sin
(
x
+
y
)
{\displaystyle A\sin(x)+B\cos(x)=C\sin(x+y)}
C
=
A
2
+
B
2
{\displaystyle C={\sqrt {A^{2}+B^{2}}}}
y
=
±
arctan
(
B
A
)
{\displaystyle y=\pm \arctan {\bigl (}{\tfrac {B}{A}}{\bigr )}}
sin
(
A
)
±
sin
(
B
)
=
2
sin
(
A
±
B
2
)
cos
(
A
∓
B
2
)
{\displaystyle \sin(A)\pm \sin(B)=2\sin \left({\frac {A\pm B}{2}}\right)\cos \left({\frac {A\mp B}{2}}\right)}
cos
(
A
)
+
cos
(
B
)
=
2
cos
sin
(
A
+
B
2
)
cos
(
A
−
B
2
)
{\displaystyle \cos(A)+\cos(B)=2\cos \sin \left({\frac {A+B}{2}}\right)\cos \left({\frac {A-B}{2}}\right)}
cos
(
A
)
−
cos
(
B
)
=
−
2
sin
(
A
+
B
2
)
sin
(
A
−
B
2
)
{\displaystyle \cos(A)-\cos(B)=-2\sin \left({\frac {A+B}{2}}\right)\sin \left({\frac {A-B}{2}}\right)}
sin
2
(
x
)
=
1
−
cos
(
2
x
)
2
{\displaystyle \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}}
cos
2
(
x
)
=
1
+
cos
(
2
x
)
2
{\displaystyle \cos ^{2}(x)={\frac {1+\cos(2x)}{2}}}
tan
2
(
x
)
=
1
−
cos
(
2
x
)
1
+
cos
(
2
x
)
{\displaystyle \tan ^{2}(x)={\frac {1-\cos(2x)}{1+\cos(2x)}}}
sin
(
−
x
)
=
−
sin
(
x
)
{\displaystyle \sin(-x)=-\sin(x)}
cos
(
−
x
)
=
cos
(
x
)
{\displaystyle \cos(-x)=\cos(x)}
tan
(
−
x
)
=
−
tan
(
x
)
{\displaystyle \tan(-x)=-\tan(x)}
csc
(
−
x
)
=
−
csc
(
x
)
{\displaystyle \csc(-x)=-\csc(x)}
sec
(
−
x
)
=
sec
(
x
)
{\displaystyle \sec(-x)=\sec(x)}
cot
(
−
x
)
=
−
cot
(
x
)
{\displaystyle \cot(-x)=-\cot(x)}
d
d
x
[
sin
(
x
)
]
=
cos
(
x
)
{\displaystyle {\frac {d}{dx}}{\big [}\sin(x){\big ]}=\cos(x)}
d
d
x
[
cos
(
x
)
]
=
−
sin
(
x
)
{\displaystyle {\frac {d}{dx}}{\big [}\cos(x){\big ]}=-\sin(x)}
d
d
x
[
tan
(
x
)
]
=
sec
2
(
x
)
{\displaystyle {\frac {d}{dx}}{\big [}\tan(x){\big ]}=\sec ^{2}(x)}
d
d
x
[
sec
(
x
)
]
=
sec
(
x
)
tan
(
x
)
{\displaystyle {\frac {d}{dx}}{\big [}\sec(x){\big ]}=\sec(x)\tan(x)}
d
d
x
[
csc
(
x
)
]
=
−
csc
(
x
)
cot
(
x
)
{\displaystyle {\frac {d}{dx}}{\big [}\csc(x){\big ]}=-\csc(x)\cot(x)}
d
d
x
[
cot
(
x
)
]
=
−
csc
2
(
x
)
{\displaystyle {\frac {d}{dx}}{\big [}\cot(x){\big ]}=-\csc ^{2}(x)}
![{\displaystyle \sin ^{2}(4x)=16{\Big [}\sin ^{2}(x)-5\sin ^{4}(x)+8\sin ^{6}(x)-4\sin ^{8}(x){\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/598b3543e3748f907f04fd36aeca7b182385ca5c)
![{\displaystyle {\frac {d}{dx}}{\big [}\sin(x){\big ]}=\cos(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0926652b30841b3f9411fa26d6ff94ef0695d9)
![{\displaystyle {\frac {d}{dx}}{\big [}\cos(x){\big ]}=-\sin(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6354b64afce92f2b1cadb5a495f5e1c88ceb25)
![{\displaystyle {\frac {d}{dx}}{\big [}\tan(x){\big ]}=\sec ^{2}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2876ba2de800a057c9e660f9ad76ac8deea0b3b)
![{\displaystyle {\frac {d}{dx}}{\big [}\sec(x){\big ]}=\sec(x)\tan(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5289ee96dbb8faf5f9bf77d6ac714b9e34d464)
![{\displaystyle {\frac {d}{dx}}{\big [}\csc(x){\big ]}=-\csc(x)\cot(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d9c7bf9123ab9d63656dff8df61248a3c4090c)
![{\displaystyle {\frac {d}{dx}}{\big [}\cot(x){\big ]}=-\csc ^{2}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efd88e218e5cfb4d074b84f856586ee308dad192)