Question

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

Answer 1

We have,

$tan\frac{x}{2}=\frac{m}{n}\Rightarrow \frac{sin\frac{x}{2}}{cos\frac{x}{2}}=\frac{m}{n}\Rightarrow sin\frac{x}{2}=mk,andcos\frac{x}{2}=nk\left(say\right)$

Now,

m sin x + n cos x

$=m.2sin\frac{x}{2}.cos\frac{x}{2}+n(co{s}^2\frac{x}{2}-si{n}^2\frac{x}{2})=2m.mk.nk+n(n^2{k}^2-m^2{k}^2)=2m^2{k}^{2}n+n{k}^2(n^2-m^2)=n{k}^2(2m^2+n^2-m^2)=n{k}^2(m^2+n^2)=n(m^2{k}^2+n^2{k}^2)=n(si{n}^2\frac{x}{2}+co{s}^2\frac{x}{2})$
=n