9.1 Identities and Proof
Identities are used for:
1. Simplifying expressions
2. Rewriting the rules of trig. functions
3. Performing numerical calculations
Phrases: “prove the identity” and “verify the identity”  mean to prove that the given
equation is an identity.
Graphical Testing - simultaneously graph the 2 functions, put 1 in y1 and 1 in y2
 If graphs are different  equation is not an identity
 If graphs appear the same  it is possible that the equation is an identity
-- however, the fact that the graphs of both sides of the equation
appear identical, does not prove that the equation is an identity
-- in window -π ≤ x ≤ π and -2 ≤ y≤ 2 graph:
Cos x =1 - x² + x4 – x6 + x8
2 24 720 40,320
Do they appear identical?
Now change window to: -2π ≤x≤ 2π – what happens?
Ex. is either of the following an identity?
3sin²x + 2cos x = 3cos²x – 2sinx
1 + cosx - cos²x = sinx + cotx
sin x
Proving Identities: -- there are no exact rules for simplifying trig. expressions or
proving identities so here are some helpful strategies:
#1. Use algebra and previously proven identities to transform 1 side of the equation into
the other.
#2. If possible – write the entire equation in terms of 1 trig. function.
#3. Express everything in terms of sine and cosine.
#4. Deal separately with each side of the equation A=B
a) 1st use identities and algebra to transform A into some expression C
b) Then use (possibly different) identities and algebra to transform B into the
same expression C
c) Conclude A=B
#5. Prove AD = BC with B≠0 and D≠0 so you can conclude
A = C
B
D
THESE TAKE PRACTICE SO DON’T HAVE A
HEARTATTACK IF YOU DON’T GET THEM
RIGHT AWAY!!!
**Proving identities is not the same as solving the equation**
Hints: #1. Whenever a squared trig. function appears --- use one of the Pythagorean
Identities.
a) sin²x  1-cos²x
b) cos²x  1 - sin²x
c) tan²x  sec²x – 1
d) csc²x  1 + cot²x
e) sec²x  tan²x + 1
f) cot²x  csc²x – 1
ex. verify that: (check to see if may be an identity)
1 + cos x - cos²x = sin x + cot x
sin x
Work on Side A only:
Step 1: Regroup terms:
(1-cos²x) + cos x
sin x
Step 2: Use Pythagorean Identity:
sin²x + cos x
sinx
 sin²x + cos x
sin x
sin x
Step 3: can combine: sin x + cot x -- which matches because of the Quotient Identity
Ex. Simplify (sec x + tan x)(1-sin x)  use proving identities and help simplify
Step 1: Reciprocal and quotient identities:
1 + sinx (1-sinx)
cos x cos x
1 + sin x (1-sinx)
cos x
(1+sin x)(1-sin x) = (1-sin²x)
cos x
cos x
= cos²x  cos x
cos x
Transform 1 side into the other side:
Ex. prove that cos x = 1 + sin x
1-sin x
cos x
**only work on one side at a time
Step 1: need to transform denominator into squared term so multiply left hand side by
cojugate
Dealing with Each side Separately
Ex. prove that secx + tan x = cos x
1-sin x
Step 1: Start with left side using the reciprocal identities
a) csc x(csc x – sin x) = cot²x
9.2 Addition and Subtraction Identities:
First Note of BEWARE:
sin(x + π/6) ≠ sin x + sin π/6
Ex. graph y1 = sin(x+π/6)
y2 = √3/2 sinx + ½cos x --- do they appear identical?
Ex. graph y1 = sin(x+5)
Y2 = sinxcos5 + cosxsin5 --- do they appear identical?
These examples of graphing lead to new identities
Addition and Subtraction identities for sine and cosine:
 Sin(x+y) = sinxcosy + cosxsiny
 Sin(x-y) = sinxcosy – cosxsiny
 Cos(x+y) = cosxcosy – sinxsiny
 Cos(x-y) = cosxcosy + sinxsiny
Ex. use the addition identities to find the exact values of sin 17π/12 and cos 17π/12
Step 1: must rewrite as a sum or difference of 2 terms for which exact values are known
such as: π/6, π/4, π/3, π/2, π
Sin 17π/12 = sin (2π/3 + 3π/4)
= sin 2π/3(cos3π/4) + cos2π/3(sin3π/4)
= √3/2 ∙-√2/2 + -½∙√2/2
= - √2(√3 + 1)
4
Ex. Subtraction Identity
Find cos(x-π/2)
= cosx(cosπ/2) + sinx(sinπ/2)
= cos x (0) + sinx (1)
= sin x
Ex. Addition Identity
Prove: 2cosxsiny = sin(x+y) – sin(x-y)
= (sinxcosy + cosxsiny) – (sinxcosy – cosxsiny)
= cosxsiny - -cosxsiny
= 2cosxsiny
Addition and Subtraction Identities for the Tangent Function
Tan (x+y) = tan x + tan y
1 – tanx tany
Tan (x-y) = tan x – tan y
1 + tanx tany
Ex. find the values of cos(x+y) and tan(x+y) if x and y are #’s such that π/2<x<π,
0<y<π/2, cos x = -2/3 and sin y = ¼. Determine in which of the following intervals x+y
lies: (0,π/2), (π/2,π), (π, 3π/2), or (3π/2, 2π)
Step 1: use Pythagorean identity and fact that sin and tan are in quadrant 1
Cofunction Identities – special cases of addition and subtraction identities
Sin x = cos (π/2 – x)
Tan x = cot (π/2 – x)
Sec x = csc( π/2 – x)
Cos x = sin (π/2 – x)
Cot x = tan (π/2 – x)
Csc x = sec (π/2 – x)
Ex. verify that sin (x – π/2) = -1
cos x
step 1: begin with the left side of equation:
sin (x-π/2) – looks almost like the cofunction identity sin(π/2 – x) but not quite
**but if factor out -1  -(x – π/2) matches identity
So: sin(x-π/2)
cos x
= sin[ -(π/2 – x)]
cos x
= -sin(π/2 –x )
cos x
= - cos x
cos x
= -1
9.3 Other Identities
Double Angle Identities – special cases of addition identities occur when 2 angles have
the same measure
Sin 2x = 2 sinxcosx
Cos 2x = cos²x - sin²x
Tan 2x = 2tanx
1-tan²x
Ex. if π<x<3/2π and cos x = -24/25 find sin 2x and cos 2x and show that 2π < 2x < 5/2π
Steps: 1) first find sin x by Pythagorean identity
sin²x = 1 - cos²x  1 – (-24/25)²  1 –576/625 = √49/625= 7/25
sin x = - 7/25 b/c per direction is in Q3
2) Sin 2x = 2sinxcosx
= 2(-7/25)(-24/25) = 336/625 = .5376
3) Cos2x = cos²x - sin²x
= (-245/25)² - (-7/25)² = .8432
4) Show that 2π<2x<5/2π
= take π<x<3/2π and multiply all by 2  2π<2x<3π
**both sin2x and cos2x are positive so 2x in Q1 and falls
2π<2x<5/2π
Ex. express the rule of the function f(x) = cos 4x in terms of powers of cos x and
constants:
Steps: use addition identity for cos 4x = cos(2x + 2x) = cosxcosy-sinxsiny
= cos2xcos2x – sin2xsin2x
** Now use double angle identity  (cos²x-sin²x)(cos²x - sin²x) – (2sinxcosx)(2sinxcosx)
= cos4x – 2sin²xcos²x + sin4x – 4sin²xcos²x
= cos4x – 2sin²xcos²x + sin²xsin²x - 4sin²xcos²x
**use Pythagorean identities
= cos4x – 2(1-cos²x)cos²x + (1-cos²x)(1-cos²x) – 4(1-cos²x)cos²x
= cos4x – 2(cos²x-cos4x) + 1 – 2cos²x + cos4x – 4(cos²x-cos4x)
= cos4x – 2cos²x + 2cos4x + 1- 2cos²x + cos4x – 4cos²x + 4cos4x
= 8cos4x – 8cos²x + 1
Forms of 2x – can be rewritten in several useful ways
Cos 2x = 1-2sin²x
Cos 2x = 2cos²x – 1
Ex. prove that 1-cos2x = 2sinx
Sinx
1-cos2x  1-(1- 2sin²x)
Sinx
sin x
= 1 -1 + 2sin²x
Sinx
2sin²x = 2sinx
sinx
Power Reducing Identities
sin²x = 1 – cos2x
2
cos²x = 1 + cos2x
2
Ex. express the function f(x) = tan²x in terms of constants and first powers with
cosine functions
f(x) = tan²x = sin²x
cos²x
= 1 – cos2x
2______
1 + cos2x
2
= 1 – cos 2x ∙
2
2
1+cos 2x
= 1-cos2x
1+cos2x
Half Angle Identities – power riding identities with x/2 in place of x is used to obtain the
half-angle identities
Sin x = ±√1-cosx
2
2
** the sign in front of the radical depends on the quadrant
Cos x = ±√1+cosx
2
2
Tan x = ±√1-cosx
2
1+cosx
Ex. find the exact value of:
a) tan 7π
12
**Problem for determining signs in the half angle formulas can be eliminated for the
tangent by using the following formulas:
Half Angle Identities for Tangent:
tan x = 1 – cosx
2
sinx
tan x = sin x
2
1+cosx
Ex. If tan x = -5/8 and 3/2π<x<π find tan x/2
Product to Sum Identities – dividing everything by 2
Sinxcosy = ½[sin(x+y) + sin(x-y)]
Sinxsiny = ½[cos(x-y) – cos(x+y)]
Cosxcosy= ½[cos(x+y) + cos(x-y)]
Cosxsiny = ½[sin(x+y) – sin(x-y)]
Sum to Product Identities – multiply everything by 2
Sinx + siny = 2sin x+y cos x-y
2
2
Sinx – siny = 2cos x+y sin x-y
2
2
Cosx + cosy = 2cos x+y cos x-y
2
2
Cosx – cosy = -2sin x+y sin x-y
2
2
9.4 Using Trig. Identities to Solve Equations
Ex. Solve cos2x + 5sinx = 3
(hint: use formula for form of cos2x 1-2sin²x)
= (1-2sin²x) + 5sinx – 3 = 0
-2sin²x + 5sinx -2 = 0
-1(2sin²x – 5sinx+2)=0
(2u-1)(u-2) = 0
Sinx = ½ sinx = 2 –(no solution b/c above -1<x<1 form chart p.580)
Sin-1x = ½ = π/6 + 2kπ and π-π/6 = 5π/6 +2kπ
Ex. solve the equation 8sinxcosx = 5
(hint: want to use double angle sin2x/cos2x/tan2x
= 4(2sinxcosx) = 5
2sinxcosx = 5/4
 replace 2sinxcosx with sin2x (double angle identity)
Sin2x = 5/4
So sin u = 5/4 where u = 2x
**but 5/4 is out of range (-1<X<1) so no solution
Ex. Find the exact solution of 2sinxcosx = ½
Sin2x = ½
(hint: double angle identity)
Ex. Find the exact solution of cos2xcos-sin2xsinx = -1
Hint: matches addition identity cos(x+y) = cosxcosy-sinxsiny
= cos(2x+x) = -1
Cos 3x = -1
Cos-1(-1) = π+ 2kπ
3x = π+2kπ
X = π/3 +2/3kπ
Ex. Find the solutions of sinx = cos x/2 where 0≤x≤2π (hint: use half angle identities)