Question

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

• A. $1$
• B. $2$
• C. $3$
• D. Infinite

Answer 1

The correct option is B $2$

$\frac{\sin \left(\lambda \alpha \right)}{\sin \alpha }-\frac{\cos \left(\lambda \alpha \right)}{\cos \alpha }=\lambda -1$
$\sin \lambda \alpha \cos \alpha -\cos \lambda \alpha \sin \alpha =(\lambda -1)\sin \alpha \cos \alpha$
$\sin (\lambda \alpha -\alpha )=\frac{(\lambda -1)}{2}\sin 2\alpha$
$so=\lambda =1:3$.