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Universidad de ValparaisoIngeniería AmbientalMatemática I
Guía 19Trigonometría: Parte 1
Prof. Juan Carlos Morgado.1
1. Demuestre las siguientes identidades trigonométricas
(a)1
tan (x)+
tan (x)
sec (x) + 1= csc (x)
(b) tan (x) + cot (x) = sec (x) csc (x)
(c) 1 + tan2 (x) = sec2 (x)
(d) sin4 (x)� cos4 (x) = 2 sin2 (x)� 1
(e)cos (x) + 1
cos (x)� 1 =1 + sec (x)
1� sec (x)
(f)sin (x)
csc (x)+cos (x)
sec (x)= 1
(g) 4 sin2 (x) cos2 (x) = 1� cos2 (2x)
(h)sec (x)
tan (x) + cot (x)= sin (x)
(i)cos (x)
1� tan (x) +sin (x)
1� cot (x) = sin (x) + cos (x)
(j) cos (x) + 2 tan (x) =�cos2 (x) + 2 sin (x)
�sec (x)
(k)sin (x) + cos (x)
sin (x)� cos (x) =sec (x) + csc (x)
sec (x)� csc (x)
(l) sec (x)� tan (x) =s1� sin (x)1 + sin (x)
(m) (1 + tan (x))2 � 2 tan (x) = sec3 (x) cos (x)
(n) 1� 2 sin2 (x) = cot (x)� tan (x)tan (x) + cot (x)
1Este material se puede obtener desde http://www.mateuv.blogspot.com/
(o)1 + cos (x)
sin (x)+
sin (x)
1 + cos (x)=
2
sin (x)
(p) 1� cos6 (x) = sin2 (x)�sin4 (x) + 3 cos2 (x)
�(q)
1
cos (x) (1 + cos (x))=tan (x)� sin (x)
sin3 (x)
(r)
stan2 (x)
1 + tan2 (x)= jsin (x)j
(s) (sin (x) + csc (x))2 = sin2 (x) + cot2 (x) + 3
(t)cos (x+ y)
cos (x� y) =1� tan (x) tan (y)1 + tan (x) tan (y)
(u) sin2 (x) + 2 cos2 (x) + cos2 (x) cot2 (x) = csc2 (x)
(v)2 tan
�x2
�1 + tan2
�x2
� = sin (x)(w) cot (x)� tan (x) = 2 cot (2x)
(x) 2 cos2 (x) =1 + sec (2x)
sec (2x)
(y)2 cos (3x)
sin (2x)+sin (2x)
cos (x)=cos (2x)
sin (x)
(z)cos (3x)
sin (x)+sin (3x)
cos (x)= 2 cot (2x)
2. Demuestre las siguientes identidades trigonométricas
(a)cos (3x)� sin (3x)cos (x) + sin (x)
= 1� 2 sin (2x)
(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)
(f) sin4 (x) + 2 sin2 (x)�1� 1
csc2 (x)
�= 1� cos4 (x)
2
(g)sin (x� y)cos (x) cos (y)
+sin (y � z)cos (y) cos (z)
+sin (z � x)cos (x) cos (z)
= 0
(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)
(j) cos6 (x)� sin6 (x) = cos (2x)�1� sin
2 (2x)
4
�
(k) (tan (x) csc (x))2 � (sin (x) sec (x))2 = 1
(l) cot2 (x)� cos2 (x) = cot2 (x) cos2 (x)
(m)1
1� sin (x) +1
1 + sin (x)= 2 sec2 (x)
(n)sin (x)� cos (x)
sin (x)+
cot2 (x)
csc (x) + 1� tan (x)
sec (x) + 1= 0
(o) cot�x2
�cot
�3x
2
��tan
�3x
2
�� 3 tan
�x2
��=
4 tan (x)
sec (x) + 2
(p) sin4 (x)�3� 2 sin2 (x)
�+ cos4 (x)
�3� 2 cos2 (x)
�= 1
(q) (tan (x) + cot (x))2 + (tan (x)� cot (x))2 =2�sin4 (x) + cos4 (x)
�sin2 (x) cos2 (x)
(r) cos (4x) = 8 cos4 (x)� 8 cos2 (x) + 1
(s)sin (3x) + sin (x)
1 + 2 cos (x) + cos (2x)= 2 cot (x) (1� cos (x))
(t) (sin (x) cos (y) + cos (x) sin (y))2 + (cos (x) cos (y)� sin (x) sin (y))2 = 1
(u) tan2��4+x
2
�� tan2
��4� x2
�= 4 tan (x) sec (x)
(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
(x)sin (6x)
sin (2x)� cos (6x)cos (2x)
= 2
(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)
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