the value of `tan9^@-tan2 7^@-tan6 3^@+tan8 1^@` is equal to

Options

A. `-1`
B. 0
C. 1
D. 4

Answer

D

Explanation

`tan9^(@)-tan27^(@)-tan63^(@)+tan81^(@)`
`=tan9^(@)-tan27^(@)-tan(90^(@)-27^(@))+tan(90^(@)-9^(@))`
`=tan9^(@)-tan27^(@)=cot27^(@)+cot9^(@)`
`=tan9^(@)+cot9^(@)-(tan27^(@)+cot27^(@))`
We know ,
`tantheta+cottheta=(1)/(sinthetacostheta)=(2)/(2sinthetacostheta)=(2)/(sin2theta)`
`thereforetan90^(@)+cot9^(@)-(tan27^(@)+tan27^(@))`
`=(2)/(sin18^(@))-(2)/(sin54^(@))=2((sin54^(@)-sin18^(@))/sin18^(@)sin54^(@))`
We know , sin C - sin D
`=2"cos"((C+D))/(2)sin((C-D)/(2))`.
`therefore2((sin54^(@)-sin18^(@))/(sin18^(@)sin54^(@)))=2*2*(cos36^(@)sin18^(@))/(sin18^(@)sin54^(@))`
`=2*2*(sin54^(@)*sin18^(@))/(sin18^(@)*sin54^(@))=4`