Necessary trigonometric identities for k-12 students [closed]

Question

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 $$\tan(\frac{\pi}{2}-x)=\cot(x)$$

Answer 1

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.

Answer 2

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."

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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.