7.3 Trigonometric identities

7.3.1 Identities for sin\sin and cos\cos

None of this document is prose (it’s all my ramblings on mathematics, and an exposition of all the things I failed to grasp explained in probably what is far too much verbosity) but if you thought the rest of it was bad (in terms of being dry) then this section is probably worse.

Here is a list of a bunch of trigonometric identities which are useful.

Theorem 7.3.1

Let θ\theta be a real number, then

sin2(θ)+cos2(θ)=1\sin^{2}(\theta)+\cos^{2}(\theta)=1 (7.6)

Proof: geometry (and hence omitted).

Theorem 7.3.2
cos(αβ)cos(α)cos(β)sin(α)sin(β)\displaystyle\cos(\alpha\mp\beta)\equiv\cos(\alpha)\cos(\beta)\mp\sin(\alpha)% \sin(\beta) (7.7)
sin(α±β)cos(α)sin(β)±sin(α)cos(β)\displaystyle\sin(\alpha\pm\beta)\equiv\cos(\alpha)\sin(\beta)\pm\sin(\alpha)% \cos(\beta) (7.8)

Proof: The \sayeasy way to prove this is using complex numbers. I was going to point out that they can be proven by using triangles, however, as previously mentioned geometry is not my thing. For a geometric proof see https://www.youtube.com/watch?v=2SlvKnlVx7U. In the "complex numbers" section of this document there’s a proof of this identity which uses the properties of complex numbers.

Theorem 7.3.3

A special case of these are the "double angle formulae" which are what we get if we set α,β=x\alpha,\beta=x in Equations 7.7 and 7.8. 55 5 These are useful for the integration of cos2(x)\cos^{2}(x) and sin2(x)\sin^{2}(x).

cos(2x)cos2(x)sin2(x)cos(2x)\equiv\cos^{2}(x)-\sin^{2}(x) (7.9)

Proof: This also serves as an easy way to remember the identities (or, to recall them if needed). This is derived from Equation 7.7 by replacing α\alpha and β\beta with xx, and then simplifying a bit:

cos(x+x)\displaystyle\cos(x+x) cos(x)cos(x)sin(x)sin(x)\displaystyle\equiv\cos(x)\cos(x)-\sin(x)\sin(x) (7.10)
cos(2x)\displaystyle\cos(2x) cos2(x)sin2(x)\displaystyle\equiv\cos^{2}(x)-\sin^{2}(x) (7.11)
Theorem 7.3.4

We also have a similar identity for sin(x)\sin(x)

sin(2x)2sin(x)cos(x).sin(2x)\equiv 2\sin(x)\cos(x). (7.12)

This is derived for 7.8 in a similar way to how the double-angle formula for cos is derived: replace α\alpha and β\beta with xx, and simplify.

sin(x+x)\displaystyle\sin(x+x) cos(x)sin(x)+sin(x)cos(x)\displaystyle\equiv\cos(x)\sin(x)+\sin(x)\cos(x) (7.13)
sin(2x)\displaystyle\sin(2x) 2cos(x)sin(x)\displaystyle\equiv 2\cos(x)\sin(x) (7.14)

7.3.2 Identities for tan(θ)\tan(\theta)

7.3.3 Secant, cosecant, cotangent and friends

These are a pain to remember. All of these functions are defined as the reciprocal of a trig function (I remember someone saying that they’re not exactly defined like this because, mumble, mumble division by zero, mumble, mumble), but the basic idea is this:

Definition 7.3.1

We define

csc(θ)=1sin(θ)\displaystyle\csc(\theta)=\frac{1}{\sin(\theta)} (7.15)
sec(θ)=1cos(θ)\displaystyle\sec(\theta)=\frac{1}{\cos(\theta)} (7.16)
cot(θ)=1tan(θ)\displaystyle\cot(\theta)=\frac{1}{\tan(\theta)} (7.17)

Here’s a helpful trick for remembering this

  1. 1.

    cosecant \mapsto sin\sin

  2. 2.

    secant \mapsto cos\cos

  3. 3.

    cotangent \mapsto tan\tan

7.3.4 Proving trigonometric identities

Example 7.3.1

Prove that

csc2(θ)(tan2(θ)sin2(θ))=tan2(θ)\csc^{2}(\theta)(\tan^{2}(\theta)-\sin^{2}(\theta))=\tan^{2}(\theta) (7.18)

This is a standard application of the general method of proving identities (as in Section 6.2); we pick one side (and my usual heuristic is to always start with the more complex side) and work toward the other side.

In this case we start with

csc2(θ)(tan2(θ)sin2(θ))=csc2(θ)tan2(θ)csc2sin2(θ)\displaystyle\csc^{2}(\theta)(\tan^{2}(\theta)-\sin^{2}(\theta))=\csc^{2}(% \theta)\tan^{2}(\theta)-\csc^{2}\sin^{2}(\theta) (7.19)

This seems like a reasonable thing to do (we must often form an \sayansatz - that is, an educated guess and try to apply this; this was my first one, but always remember the story of the mouse - as in Section 2.4 - we can always try something else if the first idea does not work), and we can proceed by unrolling some definitions

csc2(θ)(tan2(θ)sin2(θ))\displaystyle\csc^{2}(\theta)(\tan^{2}(\theta)-\sin^{2}(\theta)) =csc2(θ)tan2(θ)csc2sin2(θ)\displaystyle=\csc^{2}(\theta)\tan^{2}(\theta)-\csc^{2}\sin^{2}(\theta) (7.20)
=1sin2(θ)sin2(θ)cos2(θ)1sin2(θ)sin2(θ)\displaystyle=\frac{1}{\sin^{2}(\theta)}\frac{\sin^{2}(\theta)}{\cos^{2}(% \theta)}-\frac{1}{\sin^{2}(\theta)}\sin^{2}(\theta) (7.21)
=1cos2(θ)1\displaystyle=\frac{1}{\cos^{2}(\theta)}-1 sin2(θ)0\displaystyle\sin^{2}(\theta)\neq 0 (7.22)
=1cos2(θ)cos2(θ)\displaystyle=\frac{1-\cos^{2}(\theta)}{\cos^{2}(\theta)} (7.23)
=sin2(θ)cos2(θ)\displaystyle=\frac{\sin^{2}(\theta)}{\cos^{2}(\theta)} By the Pythagorean identity (7.24)