# What are trigonometric identities

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

In a right-angled triangle, with a hypotenuse ("hyp"), and with sides adjacent ("adj") and opposite ("opp") to the acute angle we are interested in, the six basic functions are defined as follows:

cosec = 1/sin, sec = 1/cos, cot = 1/tan.

Equations

Memorize this equation:
$$\cos^2x\,+\,\sin^2x\,=\,1$$

(it comes from Pythagoras' theorem: $\mathrm{adj}^2\,+\,\mathrm{opp}^2\,=\,\mathrm{hyp}^2$)

Divide the equation by $\cos^2x$, and rearrange terms to get:
$$\sec^2x\,-\,\tan^2x\,=\,1$$

Divide it instead by $\sin^2x$, and rearrange terms to get:
$$\mathrm{cosec}^2x\,-\,\cot^2x\,=\,1$$

Extended explanation

$$\cos2x\,=\,\cos^2x\,-\,\sin^2x$$

$$1\,+\,\cos2x\,=\,2\,\cos^2x$$

$$1\,-\,\cos2x\,=\,2\,\sin^2x$$

$$\sin{2x}\,=\,2\,\sin{x}\,\cos{x}$$

$$\sin(x\,+\,y)\,=\,\sin x\cos y\,+\,\cos x\sin y$$

$$\sin(x\,-\,y)\,=\,\sin x\cos y\,-\,\cos x\sin y$$

$$\cos(x\,+\,y)\,=\,\cos x\cos y\,-\,\sin x\sin y$$

$$\cos(x\,-\,y)\,=\,\cos x\cos y\,+\,\sin x\sin y$$​

You must learn all the equations above.

$$A\sin x\,+\,B\cos x\,=\,\sqrt{(A^2+B^2)}\sin (x\,+\,\tan^{-1}(B/A))$$

. . . . . . . . . . . . . $$=\,\sqrt{(A^2+B^2)}\cos (x\,-\,\tan^{-1}(A/B))$$

$$\sin x\,+\,\sin y\,=\,2 \sin \frac{x\,+\,y}{2} \cos \frac{x\,-\,y}{2}$$

$$\sin x\,-\,\sin y\,=\,2 \sin \frac{x\,-\,y}{2} \cos \frac{x\,+\,y}{2}$$

$$\cos x\,+\,\cos y\,=\,2 \cos \frac{x\,+\,y}{2} \cos \frac{x\,-\,y}{2}$$

$$\cos x\,-\,\cos y\,=\,-2 \sin \frac{x\,+\,y}{2} \sin \frac{x\,-\,y}{2}$$​

These last four equations are too difficult to remember , but when needed you can work them out as follows

They all have a 2, an (x+y)/2, and an (x-y)/2, and

Sum or difference of sin always has a cos and a sin, just as in sin(x±y).

Sum or difference of cos always has two coses or two sines, just as in cos(x±y).

And a sum doesn't depend on the order, so it has to have cos the difference, which also doesn't; while a difference does, so it has to have sin the difference, which also does.

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