Question

If (cos α + cos β)² + (sin α + sin β)² = $\lambda {\cos }^2\left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)$, write the value of λ.

Answer 1

(cos α + cos β)² + (sin α + sin β)² = $\lambda {\cos }^2\left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)$

Consider LHS:
(cos α + cos β)² + (sin α + sin β)²

$$\begin{gathered} ={\left[2\cos \left(\frac{\alpha +\beta }{2}\right)c\mathrm{os}\left(\frac{\alpha -\beta }{2}\right)\right]}^2+{\left[2\sin \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)\right]}^2 \\ =4{\cos }^2\left(\frac{\alpha +\beta }{2}\right){\cos }^2\left(\frac{\alpha -\beta }{2}\right)+4{\sin }^2\left(\frac{\alpha +\beta }{2}\right){\cos }^2\left(\frac{\alpha -\beta }{2}\right) \\ =4{\cos }^2\left(\frac{\alpha -\beta }{2}\right)\left[{\cos }^2\left(\frac{\alpha +\beta }{2}\right)+{\sin }^2\left(\frac{\alpha +\beta }{2}\right)\right] \\ =4{\cos }^2\left(\frac{\alpha -\beta }{2}\right) \\ =\mathrm{RHS} \end{gathered}$$