(a) Prove that `sin65^(@)+cos65^(@)=sqrt(2)cos 20^(@)`
(b) Prove that `sin47^(@)+cos77^(@)=cos17^(@)`

Explanation

(a) `sin65^(@)+cos65^(@)=sin65^(@)sin25^(@)`
`=2sin((65^(@)+25^(@))/(2))cos((65^(@)-25^(@))/(2))`
`=2sin 45^(@)cos20^(@)`
`=2(1)/sqrt(2)cos20^(@)`
`=sqrt(2)cos20^(@)`
(b) `sin47^(@)+cos77^(@)=sin47^(@)+sin13^(@)`
`=2sin((47^(@)+13^(@))/(2))cos((47^(@)-13^(@))/(2))`
`=2sin 30^(@)20^(@)`
`=2(1)/(2)cos20^(@)-cos20^(@)=0`