Section 5.2: Proving Trig. Identities ............_... _........__.-............._......._._........ _.._......_{Q
/"
..... ___
. ..........._..... _.-_...-......__ ..............._._..........­ '="'
HOMEWORK:
1,3,6 , 11 , 14,17,20,23, 24,
27, 31, 36, 41,44, 50, 56 , 62,
64, 67
~
n
'."
I:"!.
7 -,. \" ......~
•-: •'. .'
,
. .:
: - ",
: ! . ' ,.
"
'.
1
U Strqte3':J
itl ' 5eglh
~ih
-the \'¥lore com pl\ccdec/
ey..pre.§loY\ \ U 0(1'\ 10l,lJavd -the
less COf(\p\lcated ­
i.AJ
Example
1:
Prove the Identity
~-=--.\
----------------·----------------- -\Q r --------------------------- - ----------­
3_ X2
-x x - ex - 1) ex + 1)
X~ _ X2 _ (X-I)(X.tl)
X
1- x
1- x
(Xl - I)
)\ (x - I) - (X Z - I)
/
-- X
-X t
- X
t
I
I
I -X
2
Example 2: Prgve the Identity
- - -_.
__.._ - - _ .
. -\
x Z-4
O''/~-------------·-·-----··-·-·-·
"----"
- -xZ-9
x-2
5
x+3
5
(X-3)~
(~
(Xt2) - (X-3)
Xt2
Xt 3
5
5
3
Are these equal to sin(x)??? --------·--·--·----·----·-·-------CO }--·------···---------;_.-_._­
1.
[(x) = cos x
* cot x
!
2. (sin 3 x)(l
+ cot
x)
I
I
I
I
I
!
I)
J (X) ~
COSX
Got )(:::
cotx cos X
Sinx
~ (sln;,x)
~ COSx ( COS X)
srnx
(A4
CSc 1 X
-X)
:: (Sln 3x )(csc 2xj
NO ~ SIr\,x.
YES
4
Example 3: Prove the Identity
(cos x) (tan x
((osx)
(cosx)
(tanx
(SIIlX
t
t
sinx
+ cos 2 x
slnx cot x)
~(COSx ))
~
C05X
(cos >\) ( 5In X t
cos X
~cotnrnon
+ sinxcot x)
COS x)
I
denominator
~(_Sl~
COS2X)
5
p Strate3Y
#}: When
there Qre
a comrf\On
Example ~ ~
}ro.ctlo(1 !:,
find
d~rom\Y\ator.
Prove the Identity
/-~
------------------------ -- - - -(Q )-- ---------- -----------------------­
_ 1_
i-cos x
."
,- I
.
";
,\>"...
+
..'
= 2 csc 2 X
1
l+cos x
.
,
~r
'.
•
'•• ~"
-.
•
\.,
.
--- t
1- Cosx
-==)
I
tcosx
camman cienomlnqtor
It~
t
(I-casx) (I
I
-..cesx
t
cosx)
I
z
6
Example 5 : Prove the Identity
;-;,=--- ,
-,---,,-,-----,-,-"---,·- -- - - -\0
0
------------,--,------.-,---. ­
'--'"
sin x
l+cos x
Jinx
I T cos ><
~
2 (seX
I T cosx
Sinx t
2
xt
slnlx
(ITCCSX) (ITC05X)
( ITGOS)( ) ( Sin x)
t IT
( It
-:.
2 esc x
sin x
common denominator
S/n
'::"
+ l+cos x
lcosx
T
COSZ,x
d~IJ)Zx TCOSlX=V
cosx) (SlhX)
1 t 2cosx
( I fCOSX) (slnx)
-:.
2{~)
~(smx)
-
2
51nx ~ 2(5C~
2 cs;cx
7