If $ an \left(\frac{x}{2} ight)=\frac{m}{n}$, then the value of $m$ $\sin (x)+n \cos (x)$ is equal to

Question

If $	an \left(\frac{x}{2}ight)=\frac{m}{n}$, then the value of $m$ $\sin (x)+n \cos (x)$ is equal to

Aicharya · Accepted Answer

It is given that $	an \frac{x}{2}=\frac{m}{n}$, so $ \begin{aligned} m \sin (x)+n \cos (x) & =m\left(\frac{2 	an \frac{x}{2}}{1+	an ^2 \frac{x}{2}}ight)+n \frac{1-	an ^2 \frac{x}{2}}{1+	an ^2 \frac{x}{2}} \ & =m \frac{2\left(\frac{m}{n}ight)}{1+\frac{m^2}{n^2}}+n \frac{1-\left(\frac{m^2}{n^2}ight)}{1+\left(\frac{m^2}{n^2}ight)} \ & =\frac{2 m^2 n}{m^2+n^2}+\frac{n\left(n^2-m^2ight)}{m^2+n^2} \ & =\frac{n\left(2 m^2+n^2-m^2ight)}{m^2+n^2}=n \end{aligned} $

If $\tan \left(\frac{x}{2}\right)=\frac{m}{n}$, then the value of $m$ $\sin (x)+n \cos (x)$ is equal to

Solution