Question

Find the value of tan 75/tan 15

Answer 1

tan 75/tan 15 = tan(45 +30) / tan(45-30)

We know tan(A+B) = (tan A + tan B)/(1- tan A tan B)

tan (A-B) = (tan A - tan B)/(1+ tan A tan B)

tan (45 + 30 ) = (tan 45 + tan 30)/(1- tan 45 tan 30) = (1+(1/√3))(1 - (1/√3)) = (√3+1)/ (√3-1)

tan (45-30) = (tan 45 - tan 30)/(1+tan 45 tan 30) = (1-(1/√3))(1 + (1/√3)) = (√3-1)/ (√3+1)

tan 75 / tan 15 = ( (√3+1)/ (√3-1))/((√3-1)/ (√3+1)) = (√3+1)²/ (√3-1)² = (3+1+2√3)/(3+1-2√3)

= (4+2√3)/(4-2√3) = (2+√3)/(2-√3) = (2+√3) (2+√3)/ (2+√3)(2-√3)

= (4+3+4√3)/(4-3) = 7+4√3