{ ffi.
Prore the following trigonometric identities.
a)
cot2
x
cot2
bl
cl'
-rl
uJ
sitt2
x
*
sin2
x:
x sin2 x + sinl *
I
= "?"1.* . sinz x + sinz x =
cos2
x+
sin2
x=l
I , I :I
111csc'x
sec- x
+ -J"
-J"
csc'x
sec' x
l+Jan2x
cotzx+l
x+
= sinz
cos2
x=l
:tanzx
7+!an2x _sec2.x _sin2x
cotzx+l csczx cos?x =t.,nzx
cosl x ofunx ,
?
- t: -cos-x
cot ,(
:T-I=--7=sin.x-I=sinzx-(sin2x+cos2x)=-cos"x
cotx
cosx.cosx
e;'.
SeCX
i
I:
cos.x
a
tan-x
secx_I=_1" _1=7-co_s2x
x
cos- x
cos. x
_sin1
cos
f) (l + cot2 x) (1 *
(7 + co?
x(l -
cos2
cos2
x) :
x) =
x =tan2x
cos" x
1
csca
x.
sin2
x = 44'\
=
g3 2cos2a-I-1-?sin?a
2 cos2 a - 1 = 2(7 - sinz a) - 7 = 2 - 2 sinz a - I
h)
tan x
*
cot
x:
tanx+cotx=
t €*
I
=1
- 2 sinz a
x csc
sinx*c9sx=sin2x+cos2x1
1 . 1
cosx sinx cosxsinx cosxsinx- cosx sinx =secxcscx
sec
Pro,re the following trigonometric identities.
a)
secx -- cos x : sinxtanx
sinx -sin2x
sinxtanx=sinr. cos
x cos x -7-cos2x
cos x
L1 l+sinx cosx
' cosx l-sinr
I+sinx
cosx
(7+sinx)(l,-sinx)
cosx(7-sinx)
7-sinzx
cosx(7-.sinx)
1 -cos2x =secx-cosx
x cos x
cos
cos2x
cosxlT--sinr)
cosx
f -sinr
-1 , sin2 x :t+COSX
LJ
I - cosx
_sin2 x _ 7_- cos2 x _ (7+ co?x)(7- cosx)
= I + cosx
I-cosx
I-cosx
I-cosx
d)
(1 + tan x)2 + (1 - tan x)2 :2 secz x
(7 + tan xf + (7 - tan xf = 1 + 2 tan x + tan2 x + 1. - 2 tan x + tan2 x = 2(7 + tanz x) = 2 sec2 x
e)
(1 + tarrr)(1
-
tanx)
-
(1 + cotn)(1
(7 +tonxl(l -tanx)-
-
cot x1' :r*ta-n4x
tan2
x
(7 + cotx)(1-cotx) = 1 -tanz x-(7-cot2 x)
7- tala x
= cotz x - tan2, = I" - tan2 * =
tan" x
tanz x
216 Chqpter 7 Trigonometry
O Guerin, r,litcur ltcc