How can I prove \tan15^{\circ}=2-\sqrt{3} using the triple-angle formulas? \sin3x=3\sin x-4\sin^3x \cos3x=4\cos^3x-3\cos

Calvin Hess

Calvin Hess

Answered question

2022-01-27

How can I prove tan15=23 using the triple-angle formulas?
sin3x=3sinx4sin3x
cos3x=4cos3x3cosx
I know that if I substitute x=15 I can write
sin45=3sin154sin315
cos45=4cos3153cos15
What can I do next?any hints

Answer & Explanation

nebajcioz

nebajcioz

Beginner2022-01-28Added 15 answers

Let s=sin15,c=cos15 and t=sc=tan15(tan0,tan45)=(0,1) then, from your equations, since sin45=cos45 ,it follows that
s(34s2)=c(4c23)s(3(s2+c2)4s2)=c(4c23(s2+c2))
and, after dividing both sides by c3, we obtain
t(3t2)=(13t2)(t+1)(t24t+1)=0
(the factorization can be obtained by noticing that the equation on the left is satisfied by t=-1).
Can you take it from here and find the root t(0,1)?

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