1. Universidad de Valparaiso
Ingeniería Ambiental
Matemática I
Guía 19
Trigonometría: Parte 1
Prof. Juan Carlos Morgado.1
1. Demuestre las siguientes identidades trigonométricas
1 tan (x)
(a) + = csc (x)
tan (x) sec (x) + 1
(b) tan (x) + cot (x) = sec (x) csc (x)
(c) 1 + tan2 (x) = sec2 (x)
(d) sin4 (x) cos4 (x) = 2 sin2 (x) 1
cos (x) + 1 1 + sec (x)
(e) =
cos (x) 1 1 sec (x)
sin (x) cos (x)
(f) + =1
csc (x) sec (x)
(g) 4 sin2 (x) cos2 (x) = 1 cos2 (2x)
sec (x)
(h) = sin (x)
tan (x) + cot (x)
cos (x) sin (x)
(i) + = sin (x) + cos (x)
1 tan (x) 1 cot (x)
(j) cos (x) + 2 tan (x) = cos2 (x) + 2 sin (x) sec (x)
sin (x) + cos (x) sec (x) + csc (x)
(k) =
sin (x) cos (x) sec (x) csc (x)
s
1 sin (x)
(l) sec (x) tan (x) =
1 + sin (x)
2
(m) (1 + tan (x)) 2 tan (x) = sec3 (x) cos (x)
cot (x) tan (x)
(n) 1 2 sin2 (x) =
tan (x) + cot (x)
1 Este material se puede obtener desde http://www.mateuv.blogspot.com/

2. 1 + cos (x) sin (x) 2
(o) + =
sin (x) 1 + cos (x) sin (x)
(p) 1 cos6 (x) = sin2 (x) sin4 (x) + 3 cos2 (x)
1 tan (x) sin (x)
(q) =
cos (x) (1 + cos (x)) sin3 (x)
s
tan2 (x)
(r) = jsin (x)j
1 + tan2 (x)
2
(s) (sin (x) + csc (x)) = sin2 (x) + cot2 (x) + 3
cos (x + y) 1 tan (x) tan (y)
(t) =
cos (x y) 1 + tan (x) tan (y)
(u) sin2 (x) + 2 cos2 (x) + cos2 (x) cot2 (x) = csc2 (x)
x
2 tan
(v) 2 = sin (x)
x
1 + tan2
2
(w) cot (x) tan (x) = 2 cot (2x)
1 + sec (2x)
(x) 2 cos2 (x) =
sec (2x)
2 cos (3x) sin (2x) cos (2x)
(y) + =
sin (2x) cos (x) sin (x)
cos (3x) sin (3x)
(z) + = 2 cot (2x)
sin (x) cos (x)
2. Demuestre las siguientes identidades trigonométricas
cos (3x) sin (3x)
(a) =1 2 sin (2x)
cos (x) + sin (x)
(b) sin2 (5x) sin2 (2x) = sin (7x) sin (3x)
(c) (cot (x) cot (2x)) (sin (x) + sin (3x)) = 2 cos (x)
(d) sin (y) sin (x + y) + cos (y) cos (x + y) = cos (x)
(e) cos (x + y) cos (x y) = cos2 (x) sin2 (y)
1
(f) sin4 (x) + 2 sin2 (x) 1 =1 cos4 (x)
csc2 (x)
2

3. sin (x y) sin (y z) sin (z x)
(g) + + =0
cos (x) cos (y) cos (y) cos (z) cos (x) cos (z)
(h) cos (4x) cos (x) sin (4x) sin (x) = cos (3x) cos (2x) sin (3x) sin (2x)
(i) cos (x) (tan (x) + 2) (2 tan (x) + 1) = 2 sec (x) + 5 sin (x)
sin2 (2x)
(j) cos6 (x) sin6 (x) = cos (2x) 1
4
2 2
(k) (tan (x) csc (x)) (sin (x) sec (x)) = 1
(l) cot2 (x) cos2 (x) = cot2 (x) cos2 (x)
1 1
(m) + = 2 sec2 (x)
1 sin (x) 1 + sin (x)
sin (x) cos (x) cot2 (x) tan (x)
(n) + =0
sin (x) csc (x) + 1 sec (x) + 1
x 3x 3x x 4 tan (x)
(o) cot cot tan 3 tan =
2 2 2 2 sec (x) + 2
(p) sin4 (x) 3 2 sin2 (x) + cos4 (x) 3 2 cos2 (x) = 1
2 2 2 sin4 (x) + cos4 (x)
(q) (tan (x) + cot (x)) + (tan (x) cot (x)) =
sin2 (x) cos2 (x)
(r) cos (4x) = 8 cos4 (x) 8 cos2 (x) + 1
sin (3x) + sin (x)
(s) = 2 cot (x) (1 cos (x))
1 + 2 cos (x) + cos (2x)
2 2
(t) (sin (x) cos (y) + cos (x) sin (y)) + (cos (x) cos (y) sin (x) sin (y)) = 1
x x
(u) tan2 + tan2 = 4 tan (x) sec (x)
4 2 4 2
(v) cot (x) 8 cot (8x) = tan (x) + 2 tan (2x) + 4 tan (4x)
(w) sec2 (x) csc2 (y) + tan2 (x) cot2 (y) sec2 (x) cot2 (y) tan2 (x) csc2 (y) = 1
sin (6x) cos (6x)
(x) =2
sin (2x) cos (2x)
(y) 4 sin (5x) cos (3x) cos (2x) = sin (4x) + sin (6x) + sin (10x)
(z) sec2 (x) tan2 (y) tan2 (x) sec2 (y) = tan2 (y) tan2 (x)
3