L.H.S
=cos20∘+sin20∘cos20∘−sin20∘
=sin(90∘−20∘)+sin20∘sin(90∘−20∘)−sin20∘
=sin70∘+sin20∘sin70∘−sin20∘
We know that
sinC+sinD=2sin(C+D2)cos(C−D2)
sinC−sinD=2cos(C+D2)sin(C−D2)
Therefore,
=2sin(70∘+20∘2)cos(70∘−20∘2)2cos(70∘+20∘2)sin(70∘−20∘2)
=sin(90∘2)cos(50∘2)cos(90∘2)sin(50∘2)
=sin45∘cos25∘cos45∘sin25∘
=tan45cot(25∘)
=1×tan(90∘−65∘)
=tan65∘
R.H.S
Hence, proved.