सिद्ध कीजिए कि
(i) `((sin 35^(@))/(cos 55^(@)))+((cos 55^(@))/(sin 35^(@)))-2cos60^(@)=1`
(ii) `((sin47^(@))/(cos 43^(@)))+((cos 43^(@))/(sin 47^(@)))=4cos ^(2)45^(@)`
(iii) `(sin ^(2)63^(@)+sin ^(2)27^(@))/(sec ^(2)20^(@)-cot^(2)70^(@))+2sin 36^(@)sin 42^(@). sec 48^(@)sec 54^(@)=3`

Explanation

(i) L.H. S. `=((sin 35^(@))/(cos 55^(@)))+((cos 55^(@))/(sin 35^(@)))-2cos 60^(@)`
`=((cos (90^(@)-55^(@)))/(cos 55^(@)))^(2)+((cos (90^(@)-35^(@)))/(sin 35^(@)))-2cos 60^(@)`
`=((cos 55^(@))/(cos 55^(@)))+((sin 35^(@))/(sin 35^(@)))^(2)-2cos 60^(@)`
`=1^(2)+1^(2)-2xx1/2=1+1-1=1=R.H.S.`
(ii) `L.H.S.=((sin 47^(@))/(cos 43^(@)))^(2)+((cos 43^(@))/(sin 47^(@)))^(2)`
`=(sin (90^(@)-43^(@))/(cos 43^(@)))^(2)+((cos (90^(@)-47^(@)))/(sin47^(@)))=((cos 43^(@))/(cos 43^(@)))+((sin47^(@))/(sin 47^(@)))^(2)=1^(2)+1^(2)=2`
R.H.S. `=4cos ^(2)45^(@)=4xx((1)/(sqrt2))^(2)=4xx1/2=2`
अतः R.H.S. = L.H.S
(ii) L. H. S. `=(sin ^(2)63^(@)+sin^(2)(90-63)^(@))/(sec^(2)20^(@)-cso ^(2)(90-20)^(@))+2sin (90-54)^(@)sin (90-48)^(@). sec 48^(@)*sec 54^(@)`
`=(sin ^(2)63^(@)+cos ^(2)63^(@))/(sec ^(2)20^(@)-tan ^(2)20^(@))+2cos 54^(@)*cos 48^(@)*(1)/(cos 48^(@))*(1)/(cos 54^(@))`
`=1/1+2=3=` R.H.S.