Question

If an angle $\theta$ be divided into two parts such that the tangent of one part is $m$ times the tangent of the other, then their difference $\varphi$ is given by

• A. $\cos \varphi =\frac{m-1}{m+1}\cos \theta$
• B. $\sin \varphi =\frac{m-1}{m+1}\sin \theta$
• C. $\sin \varphi =\frac{m-1}{m+1}\cos \theta$
• D. $\cos \varphi =\frac{m-1}{m+1}\sin \theta$

Answer 1

The correct option is B $\sin \varphi =\frac{m-1}{m+1}\sin \theta$

Let the two parts be $A$ and $B$, then

$A+B=\theta ,A-B=\varphi$ and $\tan A=m\tan B$

$\Rightarrow \frac{\tan A}{\tan B}=\frac{m}{1}\Rightarrow \frac{\tan A-\tan B}{\tan A+\tan B}=\frac{m-1}{m+1}$ (applying componendo and dividendo)

$\Rightarrow \frac{\sin (A-B)}{\sin (A+B)}=\frac{m-1}{m+1}\Rightarrow \sin \varphi =\frac{m-1}{m+1}\sin \theta$