From Förberedande kurs i matematik 1

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-	{{NAVCONTENT_START}}	+	Let's first investigate when the equality
-	<center> [[Image:4_4_5b-1(2).gif]] </center>	+
-	{{NAVCONTENT_STOP}}	+	{{Displayed math||<math>\tan u=\tan v</math>}}
-	{{NAVCONTENT_START}}	+
-	<center> [[Image:4_4_5b-2(2).gif]] </center>	+	is satisfied. Because <math>\tan u</math> can be interpreted as the slope (gradient) of the line which makes an angle ''u'' with the positive ''x''-axis, we see that for a fixed value of <math>\tan u</math>, there are two angles ''v'' in the unit circle with this slope,
-	{{NAVCONTENT_STOP}}	+
		+	{{Displayed math||<math>v=u\qquad\text{and}\qquad v=u+\pi\,\textrm{.}</math>}}
	[[Image:4_4_5_b.gif|center]]		[[Image:4_4_5_b.gif|center]]
		+
		+	The angle ''v'' has the same slope after every half turn, so if we add multiples of
		+	<math>\pi</math> to ''u'', we will obtain all the angles ''v'' which satisfy the equality
		+
		+	{{Displayed math||<math>v=u+n\pi\,,</math>}}
		+
		+	where ''n'' is an arbitrary integer.
		+
		+	If we apply this result to the equation
		+
		+	{{Displayed math||<math>\tan x=\tan 4x</math>}}
		+
		+	we see that the solutions are given by
		+
		+	{{Displayed math||<math>4x = x+n\pi\qquad\text{(n is an arbitrary integer),}</math>}}
		+
		+	and solving for ''x'' gives
		+
		+	{{Displayed math||<math>x = \tfrac{1}{3}n\pi\qquad\text{(n is an arbitrary integer).}</math>}}

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Current revision

Let's first investigate when the equality

\displaystyle \tan u=\tan v

is satisfied. Because \displaystyle \tan u can be interpreted as the slope (gradient) of the line which makes an angle u with the positive x-axis, we see that for a fixed value of \displaystyle \tan u, there are two angles v in the unit circle with this slope,

\displaystyle v=u\qquad\text{and}\qquad v=u+\pi\,\textrm{.}

The angle v has the same slope after every half turn, so if we add multiples of \displaystyle \pi to u, we will obtain all the angles v which satisfy the equality

\displaystyle v=u+n\pi\,,

where n is an arbitrary integer.

If we apply this result to the equation

\displaystyle \tan x=\tan 4x

we see that the solutions are given by

\displaystyle 4x = x+n\pi\qquad\text{(n is an arbitrary integer),}

and solving for x gives

\displaystyle x = \tfrac{1}{3}n\pi\qquad\text{(n is an arbitrary integer).}