This document is a guide on trigonometric identities from Professor Juan Carlos Morgado of Universidad de Valparaiso. It contains over 50 trigonometric identities to prove across two sections. The identities cover basic trig functions like sin, cos, tan as well as inverse trig functions like csc and cot. Students are asked to prove each identity through algebraic manipulation.

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