If `tan 15^@=2-sqrt3`, then the value of `tan15^@ cot 75^@+tan75^@ cot 15^@` is

Options

A. 14
B. 12
C. 10
D. 8

Answer

A

Explanation

`tan 15^(@) . cot 75^(@) + tan 75^(@) . cot 15^(@)`
`= tan 15^(@). cot (90^(@) - 15^(@)) + tan (90^(@) - 15^(@)) . cot 15^(@)`
`= tan^(2) 15^(@) + cot^(2) 15^(@) " "` . . . (i) [`:. tan (90^(@) - theta) = cot theta, cot (90^(@) theta)= cot theta, cot (90^(@) - theta) = tan theta`]
Now, ` cot 15^(@) = (1)/( 2 - sqrt(3)) = (2 + sqrt(3))/( (2 - sqrt(3) ) ( 2 + sqrt(3)))`
` = 2 + sqrt(3) " " ` (`:. tan 15^(@) = 2 - sqrt(3)` )
`:. tan^(2) 15^(@) + cot^(2) 15^(@) = (2 - sqrt(3))^(2) + (2 + sqrt(3))^(2)`
` = (4 + 3 - 4 sqrt(3)) + (4 + 3 + 4 sqrt(3))`
= 14