Question

If $sin\beta$ is the geometric mean between $sin\alpha$ $cos\alpha$ then $cos2\beta$ is equal to :

• A. $2si{n}^2(\frac{\pi }{4}+\alpha )$
• B. $1-sin\left(2\alpha \right)$
• C. $2co{s}^2(\frac{\pi }{4}-\alpha )$
• D. $2si{n}^2(\frac{\pi }{4}-\alpha )$

Answer 1

The correct option is B $1-sin\left(2\alpha \right)$

Given: $sin\beta$ is the geometric mean between $sin\alpha$ and $cos\alpha$

$\therefore sin\beta =\sqrt{sin\alpha .cos\alpha }=\frac{1}{4}\sqrt{sin2\alpha }$

$Cos2\beta =1-2si{n}^2\beta =1-2sin\alpha cos\alpha =1-sin2\alpha$