Question

If $\theta +\varphi =\alpha$ and $\tan \theta =k\tan \varphi ,$ then $\sin (\theta -\varphi )=$

Answer 1

$\theta +\varphi =\alpha$ and $tan\theta =ktan\varphi$

$\frac{tan\theta }{tan\varphi }=k$

$\frac{sin\theta cos\varphi }{cos\theta sin\varphi }=\frac{k}{1}$

using componendo - Dividendo, we get

$\frac{sin\theta cos\varphi +cos\theta sin\varphi }{sin\theta cos\varphi -cos\theta sin\varphi }=\frac{k+1}{k-1}$

$\frac{sin(\theta +\varphi )}{sin(\theta -\varphi )}=\frac{k+1}{k-1}$

Hence $sin(\theta -\varphi )=\frac{k-1}{k+1}sin(\theta +\varphi )$