
1410 ELEMENTARY FUNCTIONS AND THEIR PROPERTIES
25.2.8 Trigonometric Functions of Multiple Arguments
cos 2x = 2 cos
2
x−1 = 1−2 sin
2
x,
cos 3x = −3 cos x+4 cos
3
x,
cos 4x = 1−8 cos
2
x+8 cos
4
x,
cos 5x = 5 cos x−20 cos
3
x+16 cos
5
x,
sin 2x = 2 s in x cos x,
sin 3x = 3 s in x−4 s in
3
x,
sin 4x = 4 cos x (sin x−2 sin
3
x),
sin 5x = 5 s in x−20 sin
3
x+16 sin
5
x,
cos(2nx) = 1+
n
X
k=1
(−1)
k
4
k
n
2
(n
2
−1) . . . [n
2
−(k −1)
2
]
(2k)!
sin
2k
x,
cos[(2n+1)x] = cos x
1+
n
X
k=1
(−1)
k
×
[(2n+1)
2
−1][(2n+ 1)
2
−3
2
] . . . [(2n+ 1)
2
−(2k −1)
2
]
(2k)!
sin
2k
x
,
sin(2nx) = 2n cos x
sin x+
n
X
k=1
(−1)
k
4
k
(n
2
−1)(n
2
−2
2
) . . . (n
2
−k
2
)
(2k −1)!
sin
2k− 1
x
,
sin[(2n+1)x] = (2n+1)
sin x+
n
X
k=1
(−1)
k
×
[(2n+1)
2
−1][(2n+ 1)
2
−3
2
] .