<PRE><B>Question 1198656</B>

What is the exact value of tan(15 degrees)?

The correct answer is one of the following. Which one?

A) 1- square root of 3

B) 2- 2(square root of 3)

C) 2+ square root of 3

D) 2+ 2(square root of 3)

E) 2- square root of 3
&lt;pre&gt;You can use a few identities: 
&lt;font color = blue&gt;&lt;font size = 3&gt;&lt;b&gt;1)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt; Difference of TWO angles: {{{matrix(1,5, tan(15^o), &quot;=&quot;, tan (45^o - 30^o), &quot;=&quot;, (tan (45^o) - tan (30^o))/(1 - tan (45^o)(tan (30^o))))}}} &lt;font color = blue&gt;&lt;font size = 3&gt;&lt;b&gt; &lt;==== Too COMPLEX and TIME-CONSUMING&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;

&lt;font color = blue&gt;&lt;font size = 3&gt;&lt;b&gt;2)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt; Difference of TWO angles: {{{matrix(1,5, tan(15^o), &quot;=&quot;, tan (60^o - 45^o), &quot;=&quot;, (tan (60^o) - tan (45^o))/(1 - tan (60^o)(tan (45^o))))}}} &lt;font color = blue&gt;&lt;font size = 3&gt;&lt;b&gt; &lt;==== Too COMPLEX and TIME-CONSUMING&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;

&lt;font color = blue&gt;&lt;font size = 3&gt;&lt;b&gt;3)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt; HALF-ANGLE: {{{matrix(1,5, tan(15^o), &quot;=&quot;, tan (30^o/2), &quot;=&quot;, &quot; &quot;+-sqrt((1 - cos (30^0))/(1 + cos (30^o))))}}} &lt;font color = blue&gt;&lt;font size = 3&gt;&lt;b&gt; &lt;==== EASIER to manipulate&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;
OR
&lt;font color = blue&gt;&lt;font size = 3&gt;&lt;b&gt;4)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt; HALF-ANGLE: {{{matrix(1,5, tan(15^o), &quot;=&quot;, tan (30^o/2), &quot;=&quot;, (1 - cos (30^o))/sin (30^o) )}}} &lt;font color = blue&gt;&lt;font size = 3&gt;&lt;b&gt; &lt;==== EASIEST to manipulate&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;

There&#39;re other COMBINATIONS but they are too diffucult, time-consumig, and don&#39;t make any sense when there&#39;re much easier ways to get the answer!

Using the 4&lt;sup&gt;th&lt;/sup&gt;, we get: HALF-ANGLE: {{{matrix(1,5, tan(15^o), &quot;=&quot;, tan (30^o/2), &quot;=&quot;, (1 - cos (30^o))/sin (30^o))}}} 
                                                         {{{matrix(4,2, &quot;=&quot;, (1 - sqrt(3)/2)/(1/2), &quot;=&quot;,  (1 - sqrt(3)/2) * 2, &quot;=&quot;, 2 - 2sqrt(3)/2, &quot;=&quot;, 2 - cross(2)sqrt(3)/cross(2))}}}                                              
                                                         {{{matrix(1,2, &quot;=&quot;, highlight(2 - sqrt(3)))}}}&lt;font color = blue&gt;&lt;font size = 3&gt;&lt;b&gt; CHOICE E)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;&lt;/pre&gt;</PRE>