6.1. Basic Trigonometric Identities http://www.ck12.org
The quotient identities follow from the definition of sine, cosine and tangent.
- tanθ=cossinθθ
- cotθ=cossinθθ
The odd-even identities follow from the fact that only cosine and its reciprocal secant are even and the rest of the
trigonometric functions are odd.
- sin(−θ) =−sinθ
- cos(−θ) =cosθ
- tan(−θ) =−tanθ
- cot(−θ) =−cotθ
- sec(−θ) =secθ
- csc(−θ) =−cscθ
The cofunction identities make the connection between trigonometric functions and their “co” counterparts like sine
and cosine. Graphically, all of the cofunctions are reflections and horizontal shifts of each other.
- cos(π 2 −θ)=sinθ
- sin(π 2 −θ)=cosθ
- tan(π 2 −θ)=cotθ
- cot(π 2 −θ)=tanθ
- sec(π 2 −θ)=cscθ
- csc(π 2 −θ)=secθ
Example A
If sinθ= 0 .87, find cos(θ−π 2 ).
Solution: While it is possible to use a calculator to findθ, using identities works very well too.
First you should factor out the negative from the argument. Next you should note that cosine is even and apply the
odd-even identity to discard the negative in the argument. Lastly recognize the cofunction identity.
cos(θ−π 2 )=cos(−(π 2 −θ))=cos(π 2 −θ)=sinθ= 0. 87
Example B
Use identities to simplify the following expression: tanxcotx+sinxcotsecx(xsecx)^2.
Solution: Start by rewriting the expression and replacing one or two terms that you see will cancel. In this case,
replace the cotx=tan^1 xand cancel the secant term.
=tanx·tan^1 x+sinxcotxsecx
Cancel the tangents to make a one and then use the quotient and reciprocal identities to rewrite the right part of the
expression in terms of just sines and cosines. Lastly you should cancel and simplify.
= 1 +sinx·cossinxx·cos^1 x
= 1 + 1
= 2Example C
Use identities to prove the following: cot(−β)cot(π 2 −β)sin(−β) =cos(β−π 2 ).