If cos `theta =(4)/(5) " and cos " phi =(12)/(13) " where " theta " and " phi ` both lie in the fourth quadrant find the values of
`(i) " cos " (theta + phi) " " (ii) " sin " (theta -phi) " ' (iii) " tan " (theta + phi)`

Explanation

Given : `"cos " theta =(4)/(5) " and cos " phi =(12)/(13)`
Since `theta ` lies in the fourth quadrant we have
`" cos " theta gt 0 " sin " theta lt 0 " and tan " theta 0`
Again since `phi` lies in the fourth quadrant we have
`" cos " phi gt 0 " sin " phi lt 0 " and tan " phi lt 0`
`" Now sin " theta =- sqrt(1-" cos"^(2) theta )=- sqrt(1-(16)/(l25)) = -sqrt((9)/(25)) =(-3)/(5)`
` " and tan " theta = ("sin " theta)/("cos " theta) =((-3)/(5)) xx (5)/(4) =(-3)/(4)`
` " Also sin " phi = - sqrt(1 - "cos"^(2) phi) =- sqrt(1-(144)/(169)) =- sqrt((25)/(269))=(-5)/(13)`
` " and tan " phi = ("sin " phi)/(" cos " phi) = ((-5)/(13)) xx (13)/(12)=(-5)/(12)`
` :. (i) " cos " (theta + phi) = " cos " theta " cos " phi = " sin " theta " sin " phi`
`=((4)/(5) xx (12)/(13)) -{((-3)/(5))xx((-5)/(13))}=((48)/(65) -(15)/(65)) =(33)/(65)`
` (i) " sin " (theta - phi) = " sin " theta " cos " phi " cos " theta " sin " phi`
`={((-3)/(5)) xx ((12)/(13))}-{(4)/(5) xx ((-5)/(13))}=((-36)/(65) +(20)/(65)) =(-16)/(65)`
`(ii) " tan " (theta + phi) = ("tan " theta + " tan " phi)/(1-"tan " theta " tan " phi)`
`=(((-3)/(4))+((-5)/(12)))/(1-{((-3)/(4))xx ((-5)/(12))} }=(((-7)/(6)))/((1-(5)/(16))) =((-7)/(6)) xx (16)/(11) =(-56)/(33)`