For simplicity we take,
cotx-tanx=cosx/sinx-sinx/cosx
color(white)(cotx-tanx)=(cos^2x-sin^2x)/(sinxcosx)
color(white)(cotx-tanx)=(cos2x)/(2sinxcosx)xx2
color(white)(cotx-tanx)=(2cos2x)/(sin2x)
color(white)(cotx-tanx)=2cot2x
=>color(red)( 2cot2x=cotx-tanx...to(A)
Put x=phi,2phi,4phi , into (A) we get
color(blue)((i)2cot2phi=cotphi-tanphi
color(violet)((ii)2cot4phi=cot2phi-tan2phi
color(brown)((iii)2cot8phi=cot4phi-tan4phi
We take,
LHS=tan Ø + 2 tan 2 ø + 4 tan 4 ø + 4color(brown)((2 cot 8
ø)touse(iii))
=tan Ø + 2 tan 2 ø + 4 tan 4 ø + 4(color(brown)(cot4phi-
tan4phi))
=tan Ø + 2 tan 2 ø +cancel( 4 tan 4 ø )+4cot4phi-
cancel(4tan4phi)
=tan Ø + 2 tan 2 ø + 4 cot 4 ø
=tanphi+2tan2phi+2(color(violet)((2cot4phi).....................touse(ii))
=tanphi+2tan2phi+2(color(violet)(cot2phi-tan2phi))
=tanphi+cancel(2tan2phi)+2cot2phi-cancel(2tan2phi)
=tanphi+color(blue)((2cot2phi).........................................touse(i)
=tanphi+color(blue)(cotphi-tanphi)
=cotphi
=RHS