find the exact value of cos(75) + cos(15)

Question

using sum and difference formulas for cos

Answer 1

cos(75^o) + cos(15^o) =  cos(45^o+30^o) + cos(45^o-30^o)

 

Use the cosine sum and difference of two angles identities:

 

cos(45^o)cos(30^o) - sin(45^o)sin(30^o) + cos(45^o)cos(30^o) + sin(45^o)sin(30^o)

 

= 2 cos(45^o)cos(30^o)

 

= 2*(1/√2)*(√3/2)

 

= √3 / √2 = (√6)/2

Answer 2

cosA+cosB=2cos([A+B]/2)cos([A-B]/2)

cos(75)+cos(15)=2cos([75+15]/2)cos([75-15]/2)

                        =2cos(90/2)cos(60/2)

                        =2cos(45)cos(30)

                        =2(√2/2)(√3/2)

                        =(√2)(√3/2)

                        =√6/2