How do you prove that #\frac { \cos 20^ { \circ } - \sin 20^ { \circ } } { \cos 20^ { \circ } + \sin 20^ { \circ } } = \tan 25^ { \circ }#?

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Answer 1 · 2017-07-16T19:51:54

See the proof below

We need

#sin(a+-b)=sinacos+-sinbcosa#

#cos20-sin20=sqrt2sin(a-20)#

#=sinacos20-sin20cosa#

#sina=1/sqrt2#

#cosa=1/sqrt2#

#tana=1#, #=>#, #a=45#

Therefore,

#cos20-sin20=sqrt2sin(45-20)=sqrt2sin25#

#cos20+sin20=sqrt2sin(a+20)=sinacos20+sin20cosa)#

#sina=1/sqrt2#

#cosa=1/sqrt2#

#tana=1#, #=>#, #a=45#

#cos20+sin20=sqrt2sin(45+20)=sqrt2sin65#

So,

#LHS=(cos20-sin20)/(cos20+sin20)=(sqrt2sin25)/(sqrt2sin65)=sin25/cos(90-65)=sin25/cos25=tan25#

#=RHS#

#QED#