Question

The number of real numbers $\lambda$ for which the equality
$\frac{\sin \lambda \alpha }{\sin \alpha }-\frac{\cos \lambda \alpha }{\cos \alpha }=\lambda -1,$
holds for all real $\alpha$ which are not integral multiples of $\frac{\pi }{2}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $\text{Infinite}$

Answer 1

The correct option is C $3$

$\frac{\sin \lambda \alpha \cdot \cos \alpha -\cos \lambda \alpha \cdot \sin \alpha }{\sin \alpha \cdot \cos \alpha }=\lambda -1$
$\Rightarrow \sin \left(\lambda \alpha -\alpha \right)=\frac{\lambda -1}{2}\cdot 2\sin \alpha \cdot \cos \alpha$
$\Rightarrow \sin (\lambda -1)\alpha =\frac{\lambda -1}{2}\cdot \sin 2\alpha$
To hold the equality the value of $\lambda$ will be:
$\lambda =-1,1,3$
So, there are three value of $\lambda$ for which equality holds.