Question

If $\tan (x+y)=\frac{3}{4}$ and $\tan (x-y)=\frac{8}{15}$
Prove that $\tan 2x=\frac{77}{36}$

Answer 1

We have $x+y+x-y=2x$

$\therefore \tan (x+y+x-y)$

$=\frac{\tan (x+y)+\tan (x-y)}{1-\tan (x+y)\tan (x-y)}$

Substituting the values of $\tan (x+y)=\frac{3}{4}$ and $\tan (x-y)=\frac{8}{15}$ above we get

$=\frac{\frac{3}{4}+\frac{8}{15}}{1-\frac{3}{4}\times \frac{8}{15}}$

$=\frac{\frac{45+32}{60}}{1-\frac{24}{60}}$

$=\frac{\frac{77}{60}}{\frac{60-24}{60}}$

$=\frac{77}{60}\times \frac{60}{36}=\frac{77}{36}$

Hence proved.