Which trigonometric identities are necessary for the k-12 student?

I want to make a list of trigonometruc identities for k-12 student. My list is below, please help me to improve the list by proposing
"useful formulas and necessary formulas"

  1. $\sin^2x + \cos^2x = 1$
  2. $\tan^2x+1=\sec^2x=\frac{1}{\cos^2x}$
  3. $\cot^2x+1=\operatorname{cosec}^2x=\frac{1}{\sin^2x}$
  4. $\sin(2x)=2\sin x . \cos x $
  5. $\cos(2x)=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$
  6. $\tan(2x)=\frac{2\tan x}{1-\tan^2x}$
  7. $\sin(x \pm y)=\sin x . \cos y \pm \cos x . \sin y$
  8. $\cos(x \pm y)= \cos x . \cos y - (\pm)\sin x. \sin y$
  9. $\tan(x +y)=\frac{\tan x + \tan y}{1-\tan x . \tan y}$
  10. $\tan(x -y)=\frac{\tan x - \tan y}{1+\tan x. \tan y}$
  11. $\tan x+ \cot x=\frac{1}{\sin x . \cos x}=\frac{2}{\sin 2x}$
  12. $\sin x +\sin y=2 \sin(\frac{x+y}{2}) . \cos(\frac{x-y}{2})$
  13. $\sin x-\sin y= 2\cos(\frac{x+y}{2}).\sin(\frac{x-y}{2})$
  14. $\cos x + \cos y=2\cos(\frac{x+y}{2}) . \cos(\frac{x-y}{2})$
  15. $\cos x - \cos y=-2\sin(\frac{x+y}{2}). \sin(\frac{x-y}{2})$
  16. $\sin x .\cos y=\frac{1}{2}(\sin(x+y) + \sin(x-y))$
  17. $\cos x . \cos y=\frac{1}{2}(\cos(x+y)+ \cos(x-y))$
  18. $\sin x . \sin y =-\frac{1}{2}(\cos(x+y) - \cos(x-y))$
  19. $a \sin x +b \cos x= \frac{|a|}{a}\sqrt{a^2+b^2}\sin(x+\tan^{-1}(\frac{b}{a}))$
  20. $\cos^2x=\frac{1+\cos(2x)}{2} \sin^2x=\frac{1-\cos(2x)}{2}$
  21. $\tan^2x=\frac{1-\cos(2x)}{1+\cos(2x)}$

Law of sine $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2R} \space\space\space\space A+B+C=\pi$$

Law of cosine $$\cos A=\frac{b^2+c^2-a^2}{2b.c}$$ $$\cos B=\frac{a^2+c^2-b^2}{2a.c}$$ $$\cos C=\frac{b^2+a^2-c^2}{2b.a}$$

$$\cos(\frac{\pi}{2}-x)=\sin x$$ $$\sin(\frac{\pi}{2}-x)=\cos x$$ by dividing 2 last formula $$\tan(\frac{\pi}{2}-x)=\cot(x)$$

  • 6
    $\begingroup$ Whether something is "necessary" is largely a matter of opinion, and depends on how willing you (or your students) are to derive one formula from another one. For example, if you have the formula for $\sin(a+b)$ it seems unnecessary to also have the formula for $\sin(2x)$, since the latter is easily obtained from the former. $\endgroup$
    – mweiss
    Mar 27, 2016 at 22:13
  • 6
    $\begingroup$ Yes, I was wondering whether the OP has in mind that students memorize these, or just have access to them and familiarity using them. It really goes to the question of what "necessary" means -- "necessary" for what purpose? $\endgroup$
    – mweiss
    Mar 27, 2016 at 23:09
  • 5
    $\begingroup$ @DaveLRenfro If we're being honest, I don't "remember" the sum/difference or double-angle formulas; I just derive them when needed from $e^{ix}=\cos(x)+i\sin(x)$. $\endgroup$
    – mweiss
    Mar 28, 2016 at 16:25
  • 1
    $\begingroup$ @Khosrotash Which university entrance exam? What country? This question needs more context in order to be answerable. $\endgroup$
    – mweiss
    Mar 28, 2016 at 16:26
  • 1
    $\begingroup$ I might worry that devoting much energy to memorizing hosts of such identities would be misguided... (this question is currently provocatively juxatposed with a question about mathematical concepts that pay off the most... and lists of trig identities are not it...). Either a very few, from which others can be derived, or just $e^{ix}=\cos x + i\sin x$ and stop memorizing any. Still, perhaps, if the university entrance exams are truly known to pick on such stuff, then... fine. But the genuine mathematical utility is small, if that's any part of the question. $\endgroup$ Mar 28, 2016 at 18:16

2 Answers 2


Keep in mind that the Law of Sines and Law of Cosines are not identities in the same sense that your items 1-21 are identities. Rather these two laws are geometrical facts that are particular to Euclidean geometry. Items 1-21 are analytical facts. If you define sine and cosine in terms of the exponential function, then these identities follow formally without any appeal to the geometric axioms.

  • $\begingroup$ "If you define sine and cosine in terms of the exponential function..." Yes. It appears that $e^{i \theta} = \cos \theta + i \sin \theta$ is all one needs to know in order to arrive at any of these identities. So while discussing the plethora of identities, we focus on this. $\endgroup$
    – Andrew
    Mar 29, 2016 at 13:07

I recommend you add these, though they are not equally "necessary":

Reciprocal Identities: $$\sin\theta=\frac 1{\csc\theta}, \quad \csc\theta=\frac 1{\sin\theta}$$ $$\cos\theta=\frac 1{\sec\theta}, \quad \sec\theta=\frac 1{\cos\theta}$$ $$\tan\theta=\frac 1{\cot\theta}, \quad \cot\theta=\frac 1{\tan\theta}$$

Quotient Identities: $$\tan\theta=\frac{\sin\theta}{\cos\theta}, \quad \cot\theta=\frac{\cos\theta}{\sin\theta}$$

Odd-Even Identities: $$\sin(-\theta)=-\sin\theta, \quad \csc(-\theta)=-\csc\theta$$ $$\cos(-\theta)=\cos\theta, \quad \sec(-\theta)=\sec\theta$$ $$\tan(-\theta)=-\tan\theta, \quad \cot(-\theta)=-\cot\theta$$

Period Identities: (where $k\in\Bbb Z$) $$\sin(\theta+2k\pi)=\sin\theta, \quad \csc(\theta+2k\pi)=\csc\theta$$ $$\cos(\theta+2k\pi)=\cos\theta, \quad \sec(\theta+2k\pi)=\sec\theta$$ $$\tan(\theta+k\pi)=\tan\theta, \quad \cot(\theta+k\pi)=\cot\theta$$

Half-Period Identities: $$\sin(\theta+\pi)=-\sin\theta, \quad \csc(\theta+\pi)=-\csc\theta$$ $$\cos(\theta+\pi)=-\cos\theta, \quad \sec(\theta+\pi)=-\sec\theta$$

Half-Angle Identities: $$\cos\frac{\theta}2=\pm\sqrt{\frac 12(1+\cos\theta)}$$ $$\sin\frac{\theta}2=\pm\sqrt{\frac 12(1-\cos\theta)}$$ $$\tan\frac{\theta}2=\frac{\sin\theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin\theta}$$

The most important reciprocal and quotient identities are those that define tangent, cotangent, secant, and cosecant in terms of sine and cosine. Some identities can be memorized in words, such as "Cosine and secant are even functions; the others are odd functions" and "Tangent and cotangent have periods of pi; the others have periods of two pi."

Note that your Law of Sines is missing an equals sign, and the two different parts are not separated clearly. I recommend that you use the Laws of Cosines in the non-fraction form, since they are then true even for degenerate triangles with zero-length sides.


Not the answer you're looking for? Browse other questions tagged or ask your own question.