Question

If tan $\frac{x}{2}=\frac{m}{n}$ ,then write the value of $m\sin x+n\cos x$

Answer 1

Given $\tan \frac{x}{2}=\frac{m}{n}$

Consider,

$m\sin x+n\cos x$

$=m\frac{\frac{2m}{n}}{1+\frac{m^2}{n^2}}+n\frac{1-\frac{m^2}{n^2}}{1+\frac{m^2}{n^2}}$ --------------------Since, $\sin x=\frac{2\tan \frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$ and $\cos x=\frac{1-{\tan }^2\frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$

$=\frac{2m^{2}n}{m^2+n^2}+\frac{n(n^2-m^2)}{n^2+m^2}$

$=\frac{n^3+m^{2}n}{m^2+n^2}$

$=\frac{n(n^2+m^2)}{m^2+n^2}$

$=n$