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)

3