How do you prove that #tan15=2-sqrt3#?

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Answer 1 · 2018-05-26T04:26:23

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RTP: #tan15=2-sqrt3#

#tan15 = tan(45-30)#

Recall: #tan(a-b)=(tana-tanb)/(1+tanatanb)#

#tan(45-30)=(tan45-tan30)/(1+tan45tan30)#

= #(1-1/sqrt3)/(1+1/sqrt3)#

= #((sqrt3-1)/sqrt3)/((sqrt3+1)/sqrt3)#

= #((sqrt3-1)/sqrt3)times(sqrt3/(sqrt3+1))#

= #(sqrt3-1)/(sqrt3+1)#

Rationalise the denominator
= #(sqrt3-1)/(sqrt3+1)times(sqrt3-1)/(sqrt3-1)#

= #(3-2sqrt3+1)/(3-1)#

= #(4-2sqrt3)/2#

Take out the common factor
= #(2(2-sqrt3))/2#

Simplify
= #2-sqrt3#