Double-Angle And Half-Angle Identities Worksheets With Answer Page 2
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6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS
Or, if we use equation (3) in the form
2
2
sin
x
1
cos
x
and substitute it into equation (2), we get
2
2
cos 2x
cos
x
(1
cos
x)
On simplification, this gives
2
cos 2x
2 cos
x
1
Third double-angle identity for cosine
(6)
Double-angle identities can be established for the tangent function in the same
way by starting with the sum formula for tangent (a good exercise for you).
We list the double-angle identities below for convenient reference.
DOUBLE-ANGLE IDENTITIES
sin 2x
2 sin x cos x
2
2
2
2
cos 2x
cos
x
sin
x
1
2 sin
x
2 cos
x
1
2 tan x
2 cot x
2
tan 2x
2
2
1
tan
x
cot
x
1
cot x
tan x
2
2
The identities in the second row can be solved for sin
x and cos
x to obtain
the identities
1
cos 2x
1
cos 2x
2
2
sin
x
cos
x
2
2
These are useful in calculus to transform a power form to a nonpower form.
E x p l o r e / D i s c u s s
(A) Discuss how you would show that, in general,
sin 2x
2 sin x
cos 2x
2 cos x
tan 2x
2 tan x
1
(B) Graph y
sin 2x and y
2 sin x in the same viewing window. Con-
1
2
clusion? Repeat the process for the other two statements in part A.
E X A M P L E
Identity Verification
1
2
1
tan
x
Verify the identity cos
2x
.
2
1
tan
x
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