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Lesson 5.8 Trig Identities
Due to the similarities between the primary trig ratios, there are
combinations of trigonometric functions that mean the same thing.
When a relationship is true, no matter what value θ takes, it is
called a trigonometric identity.
e.g. tan θ = sin θ / cos θ
Base Formulae:
sin 
cos 
1
csc  
sin 
1
sec  
cos 
1
cot  
tan 
tan  
Pythagorean Identities:
sin 2   cos 2   1
1  cot 2   csc 2 
tan 2   1  sec 2 
Solving Trigonometric Identities
The trick is proving that an equation is an identity. Once we do that, we
can use that identity in all sorts of applications to make our lives
simpler. To prove it, we use a left side/right side format. If we can take
each side separately, and make it equal to the same thing, then the two
sides must equal each other and our identity is proven.
There are some basic trigonometric identities that we start with, and will
then use to get others:
1. Tangent Definition: tan θ = sin θ / cos θ
Proof:
LS = tan θ
RS = sin θ
cos θ
2. Prove the first identity is true: sin2 θ + cos2 θ = 1
Using these trigonometric identities, we can prove other identities
by the same LS/RS method. Note that you can start with the LS,
RS, or work them both until they are equal.
Tips for Solving Trigonometric Identities:
1. Try to write everything in terms of sin or cos only. Use the base
formulae to rewrite anything else in these terms.
2. Start with the most complicated side, so you can try to simplify
instead of complicate.
3. Don’t be afraid to switch to the other side and meet them
somewhere in the middle.
4. Look to other identities, expanding, or factoring when you are
stuck.
5. Be patient, and work the problem. This subject takes practice
before it clicks. Most people don’t see all the steps to the answer right
away, but if you just do what you can, the answer will usually pop out
somewhere down the line.
Ex. #1 Prove cos2 θ tan θ = sin2 θ cot θ
LS = cos2 θ tan θ
RS = sin2 θ cot θ
Ex. #2 Prove that 1 + cot 2 𝜃 = csc 2 𝜃 (all angles 𝜃, where 0𝑜 ≤ 𝜃 ≤ 360𝑜 except 0𝑜 ,
180𝑜 and 360𝑜 )
Ex. #3. Prove that 1 + tan2 𝜃 = sec 2 𝜃 (all angles 𝜃, where 0𝑜 ≤ 𝜃 ≤ 360𝑜 except 90𝑜
and 270𝑜 .)
5.5.4 Proving Trig Identities by Factoring
sin 𝜃+sin2 𝜃
Eg. #4. Prove that tan 𝜃 = (cos 𝜃)(1+sin 𝜃) (for all angles 𝜃, where 0𝑜 ≤ 𝜃 ≤ 360𝑜 except
where cos 𝜃 = 0.)