Question

the value of tan75+tan15 is

Answer 1

We have tan (A + B) = tanA + tanB/ 1- tanAtanB.

Also we have tan(A-B)= tanA- tanB / 1+ tanAtanB

So tan 75= tan (45 + 30)

=tan45 + tan 30 / 1- tan45tan30

=1+ [1/root(3)] / 1- [1/root(3)]

=root (3) + 1/root (3) 1. (1)

Also tan 15 = tan(45- 30)

= tan45 tan30/1+tan45tan30

=1 [1/root(3)] / 1 + [1/root(3)]

=root (3) 1 / root(3) + 1. (2)

Tan 75 / tan15=(1) / (2)

{root(3) + 1 /root(3)- 1} / {root(3) - 1 / root(3) +1 }

Root (3) + 1 / root(3) -1 x root(3) + 1 / root (3) 1

=(root (3) + 1 )2 / (root (3) 1 )2

=(3 + 2root(3) + 1 )/(3 2root(3) +1)

=(4 + 2root(3)) / ( 4 2root(3)).

OR

Tan 75/ tan 15

=tan 75/tan (90 75)

=tan 75 / cot75

=tan 75/ (1/ tan 75)

=tan 75 x tan 75

= Root (3) + 1 / root(3) -1 x root(3) + 1 / root (3) 1

=(root (3) + 1 )2 / (root (3) 1 )2

=(3 + 2root(3) + 1 )/(3 2root(3) +1)

=(4 + 2root(3)) / ( 4 2root(3)).