Đáp án:
$\cos2\alpha=-\dfrac{17}{81}$
Giải thích các bước giải:
$\sin\dfrac{\alpha}{2}-\cos\dfrac{\alpha}{2}=\dfrac{4}{3}$
$⇔\left(\sin\dfrac{\alpha}{2}-\cos\dfrac{\alpha}{2}\right)^2=\dfrac{16}{9}$
$⇔\sin^2\dfrac{\alpha}{2}-2\sin\dfrac{\alpha}{2}\cos\dfrac{\alpha}{2}+\cos^2\dfrac{\alpha}{2}=\dfrac{16}{9}$
$⇔\left(\sin^2\dfrac{\alpha}{2}+\cos^2\dfrac{\alpha}{2}\right)-2\sin\dfrac{\alpha}{2}\cos\dfrac{\alpha}{2}=\dfrac{16}{9}$
$⇔1-\sin\alpha=\dfrac{16}{9}$
$⇔\sin\alpha=-\dfrac{7}{9}$
$⇒\cos2\alpha=1-2\sin^2\alpha=1-2.\left(-\dfrac{7}{9}\right)^2=-\dfrac{17}{81}$
Vậy $\cos2\alpha=-\dfrac{17}{81}$.
