Hello mandyn318
Welcome to Math Help Forum! Originally Posted by
mandyn318 Given that :
$\displaystyle sinB/sinA=cos(A+B)$
To prove :
$\displaystyle tan(A+B)=2tanA$
How to do this prove ? thank you very much
This looks easy, but I think it's surprisingly tricky. I have a proof that assumes $\displaystyle \cos A > 0$ and $\displaystyle \cos B>0$. I'll leave it to you to check what happens if you take the negative values of the square roots.
To make it easier to read, then, I'm going to let:$\displaystyle \sin A = a$ and $\displaystyle \sin B = b$
So:$\displaystyle \cos A = \sqrt{1-a^2}$ and $\displaystyle \cos B = \sqrt{1-b^2}$, taking the positive square roots
Therefore the original given equation:$\displaystyle \frac{\sin B}{\sin A} = \cos(A+B)$$\displaystyle =\cos A \cos B - \sin A \sin B$
can be written as:$\displaystyle \frac ba=\sqrt{(1-a^2)(1-b^2)}-ab$
$\displaystyle \Rightarrow b(1+a^2)=a\sqrt{(1-a^2)(1-b^2)}$ ...(1)
Now$\displaystyle \tan(A+B)=\sin(A+B)\cdot\frac{1}{\cos(A+B)}$
$\displaystyle =\sin (A+B)\cdot\frac{\sin A}{\sin B}$, from the original given
$\displaystyle =\frac{\sin A(\sin A \cos B + \cos A \sin B)}{\sin B}$
$\displaystyle =\frac{a(a\sqrt{1-b^2}+b\sqrt{1-a^2})}{b}$
$\displaystyle =\frac{a\Big(a\sqrt{(1-a^2)(1-b^2)}+b(1-a^2)\Big)}{b\sqrt{1-a^2}}$, multiplying top-and-bottom by $\displaystyle \sqrt{1-a^2}$
$\displaystyle =\frac{a\Big(b(1+a^2)+b(1-a^2)\Big)}{b\sqrt{1-a^2}}$ from (1)
$\displaystyle =\frac{2ab}{b\sqrt{1-a^2}}$
$\displaystyle =2\tan A$
Grandad