Prove that `(i) "sin " (7pi)/(12) " cos " (pi)/(2) - "cos " *(7pi)/(12) " sin " (pi)/(4) = (sqrt(3))/(2)`` (ii) " sin " (pi)/(4) " cos " (pi)/(2) + "cos"(pi)/(4) " sin " (pi)/(12) = (sqrt(3))/(2)` `(iii) " cos " (2pi)/(3) " cos " (pi)/(4) - " sin " (2pi)/(3) " sin " (pi)/(4) =(-(sqrt(3) +1))/(2sqrt(2))`

Explanation

`(i) " sin " (7pi)/(12) " cos " (pi)/(2) - " cos " (7pi)/(12) " sin " (pi)/(4) `
`= " sin " ((7pi)/(12)-(pi)/(4)) " " [ :' " sin x cos y - cos x sin y = sin ( x-y) "]`
`= " sin " (pi)/(3) = (sqrt(3))/(2)`
`(ii) "sin "(pi)/(4) " cos " (pi)/(12) + " cos " (pi)/(2) "sin "(pi)/(12)`
`= " sin " ((pi)/(2) + (pi)/(12)) " " [ :' "sin x cos y + cos x sin y = sin (x + y)"]`
`= " sin " (pi)/(3) = (sqrt(3))/(2)`
`(iii) " sin " (2pi)/(3) " cos " (pi)/(4) - "sin " (2pi)/(3) " sin " (pi)/(4)`
` = " cos " (pi-(pi)/(3)) " cos " (pi)/(4) - " sin " (pi-(pi)/(3)) " sin " (pi)/(4)`
` =- " cos " (pi)/(3) " cos " (pi)/(4) - " sin " (pi)/(3) " cos " (pi)/(4)`
`[ :' " sin " (pi-x) " = sin x , cos "(pi-x) =- " cos x"]`
` =(-(1)/(2) xx (1)/(sqrt(2))) - ((sqrt(3))/(2) xx (1)/(sqrt(2))) =- {((sqrt(3)+1))/(2sqrt(2))}.`