The angles A, B, C of a triangle are in the ratio 2 : 5 : 5. What is the value of tan B tan C ?

Options

A. `4 + sqrt(3)`
B. `4 + 2 sqrt(3)`
C. `7 + 4 sqrt(3)`
D. `3 + 3 sqrt(3)`

Answer

C

Explanation

Let the angles A, B and C of a triangle are 2x, 5x and 5x, respectively
So, `2x + 5x + 5x = 180^(@)`
`rArr x = (180^(@))/(12)=15^(@)`
Angles are `30^(@), 75^(@), 75^(@)`
`angle B = 75^(@) and angle C = 75^(@)`
`therefore tan B tan C = (tan 75^(@))^(2)=(tan(45^(@) + 30^(@)))^(2)`
`=((tan 45^(@) + tan 30^(@))/(1-tan 45^(@) tan 30^(@)))^(2)=((1+(1)/(sqrt(3)))/(1-(1)/(sqrt(3))))^(2)=((sqrt(3)+1)/(sqrt(3)-1))^(2)`
`=(((sqrt(3)+1)^(2))/(3-1))^(2)=(1)/(4)[3+1+2sqrt(3)]^(2)`
`=(1)/(4)[4+2sqrt(3)]^(2)=(1)/(4)[16+12+16sqrt(3)]`
`=(1)/(4)[28+16sqrt(3)]=7+4sqrt(3)`