Question

if sin beta is geometric mean between sin alpha and cos alpha , then cos2alpha

Answer 1

Answer-

$\cos 2\alpha=\sqrt{1-4\sin^4 \beta}$

Solution-

Given,

sin β is the geometric mean of sin α and cos α, i.e

$\Rightarrow \sqrt{\sin \alpha \times \cos \alpha} =\sin \beta$

$\Rightarrow (\sqrt{\sin \alpha \times \cos \alpha})^2 =(\sin \beta)^2$

$\Rightarrow \sin \alpha \times \cos \alpha=\sin^2 \beta$

$\Rightarrow 2\times \sin \alpha \times \cos \alpha=2\times \sin^2 \beta$

$\Rightarrow \sin 2\alpha=2\sin^2 \beta$

$\Rightarrow (\sin 2\alpha)^2=(2\sin^2 \beta)^2$

$\Rightarrow \sin^2 2\alpha=4\sin^4 \beta$

$\Rightarrow 1-\sin^2 2\alpha=1-4\sin^4 \beta$

$\Rightarrow \cos^2 2\alpha=1-4\sin^4 \beta$

$\Rightarrow \sqrt{\cos^2 2\alpha} =\sqrt{1-4\sin^4 \beta}$

$\Rightarrow \cos 2\alpha=\sqrt{1-4\sin^4 \beta}$