a) \(\sin {75^0} + \cos {75^0} \)
\(= \sin ({45^0} + {30^0}) + \cos ({45^0} + {30^0}) \)
\(= \sin {45^0}.\cos{30^0} + \cos {45^0}.\sin {30^0} \)\(+ \cos {45^0}.\cos{30^0} - \sin {45^0}.\sin{30^0}\)
\(= {{\sqrt 2 } \over 2}(\cos{30^0} + \sin {30^0} + \cos{30^0}\)\( - \sin {30^0}) \)
\(= {{\sqrt 2 } \over 2}.2{{\sqrt 3 } \over 2} = {{\sqrt 6 } \over 2} \)
\(\eqalign{b) \, \, & \tan {267^0} + \tan {93^0} \cr&= \tan ({267^0} - {360^0}) + \tan {93^0} \cr & = \tan ( - {93^0}) + tan{93^0} = 0 \cr} \)
\(\eqalign{c) \, \, & \sin {65^0} + \sin {55^0}\cr& = 2\sin {{{{65}^0} + {{55}^0}} \over 2}\cos {{{{65}^0} - {{55}^0}} \over 2} \cr & = 2\sin {60^0}\cos {5^0} = \sqrt 3 \cos {5^0} \cr} \)
\(\eqalign{d) \, \, & \cos {12^0} - \cos {48^0} \cr&= - 2\sin {{{{12}^0} + {{48}^0}} \over 2}\sin {{{{12}^0} - {{48}^0}} \over 2} \cr & = - 2\sin {30^0}\sin ( - {18^0}) \cr&= 2\sin {30^0}\sin {18^0} = 2.{1 \over 2}\sin {18^0} \cr&= \sin {18^0}. \cr} \)
