i.-
-
13.
Prorre the following trigonometric identities.
a)
cot2
*
x sinZ x
x+
cotz x sin2
-
c)
d)
-
+ cos2
x=7
- 1 = sinz x - 1 = sinz x - (sinz x + cos2 x) = -cos2 x
-7= cos2x.sinx.sinx
CoSX.CoS.r
x tan x
x
SeCf
a
f
secx i. = _-;_1
cos
cosx
f) (1 +
cos'x
x) (1 -
cot2
(7 + cotz xN7
s) lcos'a2 cos2
a-
-
7-cos2x sin2x ,
_ i. =
____________;_
cosz
lr)
cos2
cos. x
:
o
cos. x
1
x) = csf
x'
sin2
* =4!
t
=
:1-Zsinza
l
^f
- sinz a) - 1 = 2 - 2
7 = 2(1
sinz a
- 7 = 1 - 2 sin2 a
tanx*cotx:secxcscx
tan x + cot
14.
1
coszx.tanx -I :
1
-COS.X
cot r
-
h)
= "?"'"* . sinz x + sin2 x = co* x + sin2 x =
7+lanzx :
tanzx
cotzx+l
l+lon2x _sec2x _sin2x
cotzx+7 csczx cos?x =t..n2x
cot
1
*
I
-r
csc'x
sec'I
71
--+-+---*
csc't
set x = sin2 x
cos2
el
sinz
1,11
b)
-r
x:1
sin2
x = sinx * cosx - sin2x+cos2x
cosx sinx
cos -r stn x
77
cosxsinx
cosx sinx
sec x csc
x
Prove the following trigonometric identities.
a) secx-cosx:sinxtanx
sinx
cosx-sinzx
cosx -7-coszx
cosx -
sinxtsnx=sin*,
-
b)
l+sinr
cosr
I
-
c)
-
cos2 x
cos x
= SeC.r - cosx
cosx
l-sinrc
(7+ sinx)(1- sinx)
+ sinx
cosx
sinZr
:-:
I cosx
1 -
cosx
cosx(7
rI
+
.
- sinx)
l-
sin2
cosx(1
x
x
- sinx)
cos2
- sinx)
cosx(7
cos
x
l-.sinx
COSX
.sin2 x _ 7.- cos2 x _ (1+ cosx)(7- cosx)
I-cosx l-cosx
l-cosx
=
I
+ cosx
tan x)z :2 seczx
(1 + tan xf + (7 - tan xF = 7 + 2 tan x + tan2 x + 1 - 2 tan x + tsnZ x = 2(7 + tanz x) = 2
(1 + tan n)(1 * tan x) (1 cot x)(1 cot rc) _ 1*tan4x
d) (l + tan x)2 + (1 el
-
+
-
tan2
secz
x
x
xfl - tan x) - (7 + cot x)(1 - cot x) = 7 - tan2 x - (7 - cot2 x)
= cotz x - tan2 * = -4- - tqn2 * =' -i;^i:-
(1 + tan
,
4
I
O Gu€rin, editeur ltee
I
-
5.8
Trigonometric identities
247