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)#.