Question

Let $sin2\beta$ is the geometric mean between $sin\alpha$ and $cos\alpha$, if $cos4\beta$ is equal to $\lambda si{n}^2(\frac{\pi }{4}-\alpha )$, then $\lambda$ is

Answer 1

$sin2\beta =\sqrt{sin\alpha .cos\alpha }$
$cos4\beta =1-2si{n}^{2}2\beta =1-2sin\alpha .cos\alpha$
$={(sin\alpha -cos\alpha )}^2=2si{n}^2(\alpha -\frac{\pi }{4})$
$=2si{n}^2(\frac{\pi }{4}-\alpha )$
$\Rightarrow \lambda =2$