Trigonometric Identities
Putting the Puzzle Pieces Together
Establishing Identities
Two functions f and g are said to be identically equal if f(x) = g(x) for every value of x for which both functions are defined. Such an equation is referred to as an identity.
Establishing an Identity Use the basic trig identities to
establish the identity csc θ tan θ = sec θ
1 sin 1sec
sin cos cos
Guidelines for Establishing Identity
1. Start with the side containing the more complicated expression
2. Rewrite sums or differences of quotients as a single quotient
3. Sometimes, rewriting one side in terms of sines and cosines only will help
4. Always keep your goal in mind. Keep looking at the other side of the equation as you work
Establishing an Identity Establish the
identity2 2
2 2
2 2
sin ( ) cos ( ) ( )
( sin ) (cos )
(sin ) (cos )
1
even odd properties
Guidelines for Establishing Identities
Be careful not to handle like an equation. You cannot establish an identity by such methods as adding the same expression to each side and obtaining a true statement.
Establishing an Identity Establish the
identity
2 2
tan cot sec csc 0
sin cos 1 10
cos sin cos sin
sin cos 10
cos sin cos sin1 1
0cos sin cos sin0 0
Establishing an Identity Establish the
identity1 sin 1 sin
4 tan sec1 sin 1 sin
Solution
2 2
2
2
1 sin (1 sin ) 1 sin (1 sin )
1 sin (1 sin )
(1 2sin sin ) (1 2sin sin )
(1 sin )
4sin4 tan sec
cos4sin 1
4 tan seccos cos
Establishing an Identity Establish the
identity3
2cos sin sincot cos
sin
Solution
3
2
2 2
cos sin sin
sin sin sin
cot 1 sin
cot cos cot cos
Establishing an Identity Establish the identity
1 sin cos
cos 1 sin
Solution
2
2
1 sin 1 sin
cos 1 sin
1 sin
cos (1 sin )
cos
cos (1 sin )
cos
(1 sin )
Establishing Inverse Trig Identities
1
2Show that sin(tan )
1
Steps:
1. Let the inverse trig function be equal to
2. Look at what value you are trying to find
3. Use pythagorean identities to arrive at results
vv
v
Establishing Trig Identities
Many more examples on pgs. 478 – 479
On-line examples
Solution
1
2 2
2 2
2
tan tan
sin
sin 1tan cos sin cos
cos sec
sin tan 1 secsec
sec tan 1 11
Let v v
Trying to find
v
v
vv so
v
Establishing Inverse Trig Identities Show that
1 2cos sin 1v v
Solution
1
2 2
2 2
2 2
sin sin
cos
sin cos 1
cos 1 sin
cos 1 sin cos 1
Let v v
Trying to find
v
One Last Toughie Establish the
identity
(tan tan )(1 cot cot ) (cot cot )(1 tan tan ) 0
1 1 1 1(tan tan ) 1 (1 tan tan ) 0
tan tan tan tan
1 1 1 1tan tan tan tan 0
tan tan tan tan
0 0
FOIL
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