Question

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

Answer 1

Given:

$\tan \frac{x}{2}=\frac{m}{n}$
$$\begin{gathered} \Rightarrow \frac{\sin {\displaystyle \frac{x}{2}}}{\cos {\displaystyle \frac{x}{2}}}=\frac{m}{n} \\ \mathrm{Let}\sin \frac{x}{2}\mathrm{be}mk\mathrm{and}\cos \frac{x}{2}\mathrm{be}nk. \\ \mathrm{Now}, \\ m\sin x+n\cos x=2m\sin \frac{x}{2}\cos \frac{x}{2}+n\left({\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2}\right) \\ =2m\times mk\times nk+n\left(n^2{k}^2-m^2{k}^2\right) \end{gathered}$$

$$\begin{gathered} =2m^2{k}^{2}n+n{k}^{\mathit{2}}\left(n^2-m^2\right) \\ =n{k}^2\left(2m^2+n^2-m^2\right) \\ =n{k}^2\left(m^2+n^2\right) \\ =n\left(m^2{k}^2+n^2{k}^2\right) \\ =n\left({\sin }^2\frac{x}{2}+{\cos }^2\frac{x}{2}\right) \\ =n\left(1\right) \\ \therefore m\sin x+n\cos x=n \end{gathered}$$