+yry$-
ffi,
r{n
fi
fi.
Use reciprocal
identitiesto rew'it"
Sffi
rewrite secx and tanx in termE of
The ausvser is n. start by using reciprocal and ratio identities to
rewrite the expression
,irr" urra cosine (we dared youj. N"rt, use-your Lnowledge of fractions to you can' Here's what it
where
cancel
and
as a division p.out"m. irre", *"itipryuy the reciprocal
should look like:
51g26'J
1
nncu
'
'1
cos, -sinx. 1 . sinx = sinx::f :=+ : *"6 r-J='Q9s+'=r
'tanr:itrI-.cos,(cosx-----cosxsinxrn{x#x
sinx.secx
cos
"1.
rf
mplily cotx'sec-r.
simp
#itr-*Tt-ffi
3.
Simp Iify sin3x' csc'x + lanx ' cosx'
Hft,ffi
2"
Simplify
ffiffi
Simplify
cotx'sinx'tanx'
e>rpressionr
Pythagorean identities are extremely helpful for simplifying complextrig
(turn
to Chapter 6
circle
a
uuit
on
triangles
right
those
from
derived
are
ThesJidentities
sino
a
triangle,
of
= the y
for a review if you need to). Remember that coso = the x leg
the
is
1'
Given
circle
unit
that
on
Ieg of a trianqle, and the hypotenuse of the triangle
two
The
other
identily.
Pythagorgan
get
first
the
i".t tn"t bgzl* r;"r, = f,ypoienusez, we
you
to
see how this
want
il
Dummies
For
(ir,ect
otn'Pri-Calculus
*" a".i""jfrom"that
that have
works!). These identities are especiallyl-rqlpful when simplifying expressions
identithe
Pythagorean
are
a term that has been squared (sin', cos2, and so on). Here
ties (and some derivatives):
or cos'x = 1 - sin2x
or tan2x=sec'x-1
or cot2x = csc?- 1
sin2x* cos'x=1
tanzx+ 1 = sec2x
1+ cot2x= csc2x
Simplify (secx * tanx)(1
-
'or'
sin'x = 7- c0x.2x
= sec2x -larfx
or
7
or
1=
csc2x- cot2x
The steps looh like this:
sinx)(cosx)-
(secx
+
tarixXl - sinxXcosx)
,i
coszx. Start by changirig everything to sine
and cosine using the reciprocal and.ratio
identities from the previous section. Then
add the resulting fractions (the common
denominator is cosine).and cancel the
cosine in the numerator and den-ominator.
This leaves you witfr two terms that you
can FOIL. Recognize this iast term as a
Pythagorean identity? We hoped you mightl
Substitute it in and you have your answer'
1 * sirl x ] (1 - sin x)(cos n) :
[\ cosx
cosx /
'
( t + sinx ) (1 - sinx)(cosx)
:
\ COSX /
t-
si]r \\ G- sin x)(poEi)
\ r.sfx )
[
1
+
=
.
(1+ sinx)(1
1- sin'x
cos'x
:
- sinx)
6"
Simplifvl- sinx'tanx
secx
Simplify sinx. cotzx + sirx.
E"
Simplify (sin2x- lxtan'zx + 1).
ffi
ffiffi
Simpl ify-
!={(tanx
slnx
+
cotx).
-
W#FA