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Trigonometric Identities
We have seen several identities involving trigonometric functions. These are often called trigonometric identities. The rest of this page and the beginning of the next page list the trigonometric identities that we’ve encountered.
Pythagorean identity Definition of tan, sec, csc, cot
cos(x)^2 + sin(x)^2 = 1 tan(x) = (^) cos(sin(xx))
sec(x) = (^) cos(^1 x)
csc(x) = (^) sin(^1 x)
cot(x) = cos(sin(xx))
Half rotation identities Quarter rotation identities
cos(x + π) = − cos(x) sin(x + π 2 ) = cos(x) sin(x + π) = − sin(x) cos(x − π 2 ) = sin(x)
Sum formulas Double angle formulas
cos(x + y) = cos(x) cos(y) − sin(x) sin(y) cos(2x) = cos(x)^2 − sin(x)^2 sin(x + y) = sin(x) cos(y) + cos(x) sin(y) sin(2x) = 2 sin(x) cos(x)
Even identities Odd identities
cos(−x) = cos(x) sin(−x) = − sin(x) sec(−x) = sec(x) tan(−x) = − tan(x) csc(−x) = − csc(x) cot(−x) = − cot(x)
Period identities Cofunction identities
cos(x + 2π) = cos(x) sin(π 2 − x) = cos(x) sin(x + 2π) = sin(x) cos(π 2 − x) = sin(x) sec(x + 2π) = sec(x) csc(x + 2π) = csc(x) tan(π 2 − x) = cot(x) cot(π 2 − x) = tan(x) tan(x + π) = tan(x) cot(x + π) = cot(x) sec(π 2 − x) = csc(x) csc(π 2 − x) = sec(x)
There are 31 trigonometric identities listed above. We’ve seen explanations for why the first 27 are true. In the remainder of this chapter — in Claims 21 and 22 — we’ll see why two of the remaining 4 identities are true. You’ll check that the other two are true for homework. Also for homework you’ll be asked to explain why the “difference formulas” for sine and cosine are true.
Claim (21). If x ∈ R, then tan(π 2 − x) = cot(x).
Proof: We know from the definition of tangent that
tan
(π
2
− x
sin(π 2 − x) cos(π 2 − x)
On page 276 in the chapter “Cosecant, Secant, and Cotangent”, we had checked that the first two cofunction identities from the top of this page are true. Now we can use those two identities to add to the equation above:
tan
(π
2
− x
sin(π 2 − x) cos(π 2 − x)
=
cos(x) sin(x)
Exercises
Write a proof for each of the following four claims. Claims A and B are the last of the six cofunction identities listed in this chapter. You might want to use the definitions of sec and csc along with the cofunction identities for sin and cos. The proofs will be somewhat similar to the proofs of Claims 21 and 22. Claims C and D are called difference formulas. Some books list them as important identities. You don’t have to be too familiar with these identities though as they are really just a combination of the more important sum formulas and even/odd identities for cos and sin. (Remember that x − y = x + (−y).)
Claim A. If x ∈ R, then sec(π 2 − x) = csc(x).
Claim B. If x ∈ R, then csc(π 2 − x) = sec(x).
Claim C. If x, y ∈ R, then cos(x − y) = cos(x) cos(y) + sin(x) sin(y).
Claim D. If x, y ∈ R, then sin(x − y) = sin(x) cos(y) − cos(x) sin(y).
