what is the exact value of tan4pi5tanpi30 divided by 1tan4pi

what is the exact value of tan(4pi/5)+tan(pi/30) divided by 1-tan(4pi/5)tan(pi/30)

Solution

given [tan(4pi/5)+tan(pi/30)] / [ 1-tan(4pi/5)tan(pi/30) ]

but tan(A+B) = [ tanA + tanB ] / [ 1- tanA. tanB ]

so here A= 4pi/5 and B = pi/30

so [tan(4pi/5)+tan(pi/30)] / [ 1-tan(4pi/5)tan(pi/30) ] = tan (4pi/5 + pi/30)

= tan( (24pi +pi)/30 )

= tan(25pi/30)

= tan(5pi/6)

= tan(pi -pi/6)

= -tan(pi/6)

= - 1 / sqrt(3)