Question

how get the answer with explanation??​

Answer 1

Solution:-

Given:-

$\to\rm \: \cos( \alpha + \beta ) = \frac{4}{5}$

$\to \rm\sin( \alpha - \beta ) = \frac{5}{13}$

To find

$\rm \: value \: \: of \: \: \tan(2 \alpha )$

We can write as

$\tan( \alpha + \beta + \alpha - \beta )$

Take

$\to\rm \: \cos( \alpha + \beta ) = \frac{4}{5} = \frac{b}{h}$

$\rm \: b = 4 \: \: \: h \: = 5 \: \: and \: p = x$

Using pythagoras theorem to find

$\rm \: {h}^{2} = {b}^{2} + {p}^{2}$

$\rm \: 25 = 16 + {p}^{2}$

$\rm \: p {}^{2} = 9$

$\rm \: p = 3$

We get

$\rm \: b = 4 \: \: \: h \: = 5 \: \: and \: p = 3$

So

$\rm \tan( \alpha + \beta ) = \frac{3}{4} = \frac{p}{b}$

Now take

$\rm\sin( \alpha - \beta ) = \frac{5}{13} = \frac{p}{h}$

$\rm \: p \: = 5 \: \: \: h = 13 \: \: \: and \: \: b = x$

Using pythagoras theorem to find

$\rm{h}^{2} = {p}^{2} + {b}^{2}$

$\rm {13}^{2} = {5}^{2} + {b}^{2}$

$\rm \: 169 = 25 + {b}^{2}$

$\rm {b}^{2} = 144$

$\rm \: b = 12$

$\rm \: p \: = 5 \: \: \: h = 13 \: \: \: and \: \: b = 12$

Now

$\rm \: \tan( \alpha - \beta ) = \frac{5}{12} = \frac{p}{b}$

so

$\tan( \alpha + \beta + \alpha - \beta )$

Formula

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

Put the value on formual

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

Taking lcm

$\rm \: \frac{ \frac{14}{12} }{1 - \frac{15}{48} }$

$\rm \: \frac{ \frac{14}{12} }{ \frac{33}{48} }$

$\rm \: \frac{14}{12} \times \frac{48}{33}$

$\rm \: \frac{14 \times 4}{33}$

$\to \rm \: \frac{56}{33}$

Answer:- option A is correct