If tan(a+b)=1/3 &tan(a-b)=2/5 then

Question

prove that tan2a=11/13 & tan2b=-1/17?

Answer 1

tan (a +b) = 1 /3

tan (a -b) = 2 /5

Rewriting tan(2a) as tan (a+ b +a-b)

Using the formula, tan (a + b) = tana + tan b/1- tana tanb

tan (a +b+a-b) = tan (a+b)+ tan(a -b) /1- tan(a+b)tan(a-b)

tan 2a = (1/3)+(2/5) / 1 -(1 /3)(2/5)

           = (5+6)/15 / 1 -(2/15)= 11/13.

Hence tan 2a = 11 /13.

Using the formula ,tan(a-b) = tan a- tan b /1 +tana tanb

Rewriting tha (2b) as tan (a+b -( a-b))

tan(a+b)- tan(a-b)/1 +tan(a+b)tan(a-b)

tan (2b)= (1/3) - (2/5) /1+(1/3)(2/5)  = (-1/15) /(17/15) = -1/17.

Hence tan(2b) = - 1/17.

Hence tan (2a)=11 /13 , and tan(2b) = -1 /17.