Question

If tan (a + b) = 5/12 and cot (a - b) =4/ 3. then tan 2b is equal to (1)-16/63 (2) 12/35 (3) -9/28​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm \: tan(a + b) = \dfrac{5}{12}$

and

$\rm \: cot(a - b) = \dfrac{4}{3}$

$\rm\implies \:\rm \: tan(a - b) = \dfrac{3}{4}$

Now, Consider

$\rm \: tan2b$

can be rewritten as

$\rm \: = \: tan(b + b)$

$\rm \: = \: tan(b + b + a - a)$

can be rewritten as

$\rm \: = \:tan[(a + b) - (a - b)]$

$\rm \: = \:\dfrac{tan(a + b) - tan(a - b)}{1 + tan(a + b) \: tan(a - b)}$

$\rm \: = \:\dfrac{\dfrac{5}{12} - \dfrac{3}{4} }{1 + \dfrac{5}{12} \times \dfrac{3}{4} }$

$\rm \: = \:\dfrac{\dfrac{20 - 36}{48} }{1 + \dfrac{15}{48}}$

$\rm \: = \:\dfrac{\dfrac{- 16}{48} }{ \dfrac{48 + 15}{48}}$

$\rm \: = \: - \: \dfrac{16}{63}$

Hence,

$\rm\implies \:\boxed{\sf{  \: \: \rm \: tan2b \: = \: - \: \dfrac{16}{63} \: \: \: }}$

Hence, Option (1) is correct.

$\rule{190pt}{2pt}$

Formula used :-

$\boxed{\sf{  \:\rm \: tan(x - y) = \frac{tanx - tany}{1 + tanx \: tany} \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\boxed{\sf{  \:\rm \: sin(x + y) = sinxcosy + sinycosx \: \: }}$

$\boxed{\sf{  \:\rm \: sin(x - y) = sinxcosy - sinycosx \: \: }}$

$\boxed{\sf{  \:\rm \: cos(x + y) = cosxcosy - sinxsiny \: \: }}$

$\boxed{\sf{  \:\rm \: cos(x - y) = cosxcosy + sinxsiny \: \: }}$

$\boxed{\sf{  \:\rm \: tan(x + y) = \frac{tanx + tany}{1 - tanx \: tany} \: \: }}$