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MCR3U
Trigonometric Identities
Use the above ratios to determine:
(a)
sin 
cos 
(b) sin2θ + cos2 θ
A trigonometric identity is any mathematical equation with trigonometric expressions that is true for all
values of the angle (variable)
Fundamental Trigonometric Identities
sin x
cos x
1) Quotient Identity
tan x 
2) Pythagorean Identities
sin2 x + cos2 x = 1
sin2 x = 1 – cos2 x
cos2x = 1 – sin2 x
1 + tan2 x = sec2 x
1 + cot2 x = csc2 x
3) Reciprocal Identities
csc x 
cos x
1
1
, sec x 
, cot x 
sin x
cos x
sin x
2
** It is not always obvious that both sides of a trigonometric expression are equal. To prove that it is
an identity, a proof that shows that both sides of the expression are equal is required.
To prove that a given expression is an identity, follow the steps:
 Separate the two sides of the equation using L.S. R.S. method
 Simplify the more complicated side until it is identical to the other side or simplify both sides
into the same expression
 Strategies to use:
(i)
express all tangent & reciprocal functions in terms of sine and cosine
(ii)
apply a Pythagorean identity if required
(iii) factor or find a common denominator where necessary
x
y
y
, cos  = , tan  =
to prove the
r
r
x
1 + tan2  = sec 2 
Example # 1 – Use the definitions sin  =
identity
Left Side
Right Side
Example # 2 – Simplify each expression using the trigonometric identities
a) (cos x) (tan x)
b) cos2 x + sin2 x
c) (sin x) (cos x)
1 – sin2 x
3
Example # 3 – Prove each identity
b) tan x 
a) (tan x) (cos x) = sin x
c)
1
1

tan x sin x cos x
tan 2 x
 sin 2 x
2
1  tan x
LS
Example # 4 – Factor each expression
a) 1 – sin2 
b) sin  – sin2 
d) sin4  – cos4 
c) sin2  – 2 sin  + 1