Simplify $(tan((5pi)/14)+tan(pi/14))/(1-tan((5pi)/14)tan(pi/14))$ ?

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Answer 1 · 2017-03-29T15:58:41

$(tan((5pi)/14)+tan(pi/14))/(1-tan((5pi)/14)tan(pi/14))=tan((3pi)/7)$

Perhaps questioner means $(tan((5pi)/14)+tan(pi/14))/(1-tan((5pi)/14)tan(pi/14))$

For this we can use the identity

$(tanA+tanB)/(1-tanAtanB)=tan(A+B)$

Here putting $A=(5pi)/14$ and $B=pi/14$ , we get

$(tan((5pi)/14)+tan(pi/14))/(1-tan((5pi)/14)tan(pi/14))$

and hence

$(tan((5pi)/14)+tan(pi/14))/(1-tan((5pi)/14)tan(pi/14))=tan((5pi)/14+pi/14)$

= $tan((6pi)/14)$

= $tan((3pi)/7)$

Simplify (tan((5pi)/14)+tan(pi/14))/(1-tan((5pi)/14)tan(pi/14))(tan((5pi)/14)+tan(pi/14))/(1-tan((5pi)/14)tan(pi/14))?