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VERIFYING TRIGONOMETRIC
IDENTITIES
Verifying Trigonometric Identities ò Although there are similarities, verifying that a trigonometric equation is an identity is quite different from solving an equation. There is no well-defined set of rules to follow in verifying trigonometric identities, and the process is best learned by practice.
Verifying Trigonometric Identities
ò Verifying trigonometric identities is a useful process if you need to convert a trigonometric expression into a form that is more useful algebraically.
ò When you verify an identity, you cannot assume that the two sides of the equation are equal because you are trying to verify that they are equal.
ò As a result, when verifying identities, you cannot use operations such as adding the same quantity to each side of the equation or cross multiplication.
Example – Verifying a Trigonometric Identity
ò Verify the identity (sec2 θ – 1) / (sec2 θ ) = sin2 θ.
ò Solution:
ò The left side is more complicated, so start with it. Pythagorean identity
Simplify.
Reciprocal identity
Quotient identity
Simplify.
Example – Verifying a Trigonometric Identity
RS = 11− sinθ
+1
1+ sinθ=
1+ sinθ1− sinθ( ) 1+ sinθ( )
+1− sinθ
1− sinθ( ) 1+ sinθ( )=
1+ sinθ +1− sinθ1− sinθ( ) 1+ sinθ( )
= 21− sin2θ
=2
cos2θ= 2sec2θ
Example – Verifying a Trigonometric Identity
LS = tan2 x +1( ) cos2 x −1( ) =
sec2 x( ) −sin2 x( ) = 1cos2 x −sin2 x( ) =
−sin2 xcos2θ
= − tan2 x
Example – Verifying a Trigonometric Identity
LS = tan x + cot x =
sin xcos x
+cos xsin x
=sin xsin xsin xcos x
+cos xcos xsin xcos x
=
sin2 x + cos2 xsin xcos x
=1
sin xcos x=
1sin x
⋅1cos x
= csc xsec x
Example – Verifying a Trigonometric Identity
RS = cos x1− sin x
⋅1+ sin x1+ sin x
=cos x 1+ sin x( )1− sin x( ) 1+ sin x( )
=
cos x 1+ sin x( )1− sin2 x
=cos x 1+ sin x( )
cos2 x=
1+ sin x( )cos x
=
1cos x
+sin xcos x
= sec x + tan x
Example – Working with Each Side Separately
LS = cot2 x
1+ csc x=csc2 x −1csc x +1
=csc x −1( ) csc x +1( )
csc x +1= csc x −1
RS = 1− sin xsin x
=1sin x
−sin xsin x
= csc x −1