Trigonometric Identities
Pythagorean Identities
Ratio Identities
Reciprocal Identities
sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
cot2 θ + 1 = csc2 θ
Odd/Even Identities
sin (–x) = –sin x
csc (–x) = –csc x
cos (–x) = cos x
sec (–x) = sec x
tan (–x) = –tan x
cot (–x) = –cot x
Sum/Difference Identities
Double Angle Identities
Half Angle Identities
or
or
or
Product to Sum Identities
or
Sum to Product Identities
Trigonometric Identities Practice Problems
1. Simplify the trigonometric expression (sin x + cos x)2 + (sin x - cos x)2
2. Verify
the identity cos x * tan x = sin x
3. Verify the identity cot x * sec x * sin x = 1
4. Verify the identity [ cot x - tan x ] / [sin x * cos] = csc2x - sec2x
1. Simplify the trigonometric expression (sin x + cos x)2 + (sin x - cos x)2
First expand the squares.
(sin x + cos x)2 + (sin x - cos x)2
= (sin2x + cos2x + 2cos x sin x) +(sin2x + cos2x - 2cos x sin x)
Group like terms.
= 2 sin2x + 2 cos2x
Factor 2 out
= 2 (sin2x + cos2x)
Use the identity sin2x + cos2x = 1 to simplify the above expression.
=2
2. Verify the identity cos x * tan x = sin x
We start with the left side and transform it into sin x. Use the identity tan x = sin x / cos x
in the left side.
cos x * tan x = cos x * (sin x / cos x) = sin x
3. Verify the identity cot x * sec x * sin x = 1
Use the identities cot x = cos x / sin x and sec x = 1/ cos x in the left side.
cot x * sec x * sin x = (cos x / sin x) * (1/ cos x) * sin x
Simplify to obtain.
(cos x / sin x) * (1/ cos x) * sin x = 1
4. Verify the identity [ cot x - tan x ] / [sin x * cos] = csc2x - sec2x
We use the identities cot x = cos x / sin x and tan x = sin x / cos x to transform the left side
as follows.
[ cot x - tan x ] / [sin x * cos] = [cos x / sin x - sin x / cos x] / [sin x * cos]
Rewrite the upper part of the above with a common denominator .
= [cos 2x / sin x * cos x- sin 2x / cos x * sin x] / [sin x * cos]
= [cos 2x - sin 2x] / [sin x * cos]2 (expression 1)
We now transform the right side using the identities csc x = 1 / sin x and sec x = 1 / cos x.
csc2x - sec2x = (1/sin x)2 - (1/cos x)<2
We now rewrite the above expression with a common denominator = [ cos2x - sin2x ] / [sin
x * cos]2 (expression 2)
We have transformed the left side to expression 1 and the right side to expression 2. These
two expressions are equal. We have verified the given identity.