Prove that tan∅/1-tan∅-cot∅/1-cot∅=cos∅+sin∅/cos∅-sin∅?

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Answer 1 · 2018-02-04T05:11:11

Kindly refer to a Proof in the Explanation.

We have, $tanphi/(1-tanphi)-cotphi/(1-cotphi)$ ,

$=tanphi/(1-tanphi)-(1/tanphi)/{(1-1/tanphi)}$ ,

$=tanphi/(1-tanphi)-(1/tanphi)/{(tanphi-1)/tanphi}$ ,

$=tanphi/(1-tanphi)-1/(tanphi-1)$ ,

$=tanphi/(1-tanphi)+1/(1-tanphi)$ ,

$=(tanphi+1)/(1-tanphi)$ ,

$=(sinphi/cosphi+1)/(1-sinphi/cosphi)$ ,

$={(sinphi+cosphi)/cosphi}-:{(cosphi-sinphi)/cosphi}$ ,

$=(cosphi+sinphi)/(cosphi-sinphi)$ .

BONUS :

Since, $tan(pi/4+phi)={tan(pi/4)+tanphi}/{1-tan(pi/4)tanphi}$ ,

$=(1+tanphi)/(1-tanphi)$ , we have,

$tanphi/(1-tanphi)-cotphi/(1-cotphi)=(cosphi+sinphi)/(cosphi-sinphi)=tan(pi/4+phi)$ .