Find the values of following . (a) `(tan^(2)37(1)/(7)^(@)+1)/(tan^(2)37(1)/(2)^(@)-1)` (b) `(3 tan^(2)5^(@)-1)/(3tan5^(@)-tan^(3)5^(@))`

Answer

(a) `2(sqrt(6)+sqrt(2))` (b) `2(2+sqrt(3))`
(c) `-(sqrt(6)+sqrt(2))` (d) `-(2+sqrt(3))`

Explanation

(a) `(1+tan^(2)7(1)/(2)^(@))/(tan7(1)/(2)^(@))=s(1+tan^(2)7(1)/(2)^(@))/(2tan7(1)/(2)^(@))`
`=(2)/(sin15^(@))`
`=((2)/sqrt(3-1))/(2sqrt(2))`
`=2sqrt(2)(sqrt(3)+1)`
`=2(sqrt(6)+sqrt(2))`
(b) `(tan^(2)82(1)/(2)^(@)-1)/(tan82(1)/(2)^(@))=-2(1-tan^(2)82(1)/(2)^(@))/(2tan82(1)/(2)^(@))`
`=(-2)/(tan165^(@))`
`=(2)/(tan15^(@))`
`=(2)/(2-sqrt(3))`
`=2(2+sqrt(3))`
(c) `(tan^(2)37(1)/(2)^(@)+1)/(tan^(2)37(1)/(2)^(@)-1)=(-1)/(cos 75^(@))`
`=(-2sqrt(2))/(sqrt(3)-1)`
`=-sqrt(2)(sqrt(3)+1)`
`=-(sqrt(6)+sqrt(2))`
(d) `(3tan^(2)5^(@)-1)/(3tan5^(@)-tan^(3)5^(@))=(-1)/(tan15^(@))`
`(-1)/(2-sqrt(3))`
`=-(2+sqrt(3))`