Question

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

 

 

Answer 1

Given: \[\tan\frac{x}{2} = \frac{m}{n}\]

\[\Rightarrow \frac{\sin\frac{x}{2}}{\cos\frac{x}{2}} = \frac{m}{n}\]
\[\text{ Let }  \sin\frac{x}{2} \text{ be mk and } \cos\frac{x}{2} \text{ be nk }  . \]
\[\text{ Now } , \]
\[m\text{ sin } x + n\text{ cos } x = 2m \sin\frac{x}{2}\cos\frac{x}{2} + n\left( \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} \right)\]
\[ = 2m \times mk \times nk + n\left( n^2 k^2 - m^2 k^2 \right)\]

\[= 2 m^2 k^2 n + n k^2 \left( n^2 - m^2 \right)\]
\[ = n k^2 \left( 2 m^2 + n^2 - m^2 \right)\]
\[ = n k^2 \left( m^2 + n^2 \right)\]
\[ = n\left( m^2 k^2 + n^2 k^2 \right)\]
\[ = n\left( \sin^2 \frac{x}{2} + \cos^2 \frac{x}{2} \right)\]
\[ = n\left( 1 \right)\]
\[ \therefore m\text{ sin } x + n\text{ cos } x = n\]