Cecil was tasked to calculate $\cos\frac{x}{2}$ without a calculator. Additionally, he knew that: $$\sinx=-\frac{\sqrt{56}}{9},\ \mathrm{ and }\ x\in\left(\frac32\pi,2\pi\right).$$
In which step of his solution did Cecil make a mistake?
Cecil´s solution:
(1) Cecil claimed that $\cos^2x=1-\sin^2x$ and therefore it holds: $$\cos^2x=1-\frac{56}{81}$$
(2) From that equality, he expressed $\cos x$: $$\cosx=\frac59$$
(3) Further, Cecil claimed that $$\left|\cos\frac{x}{2}\right|=\sqrt{\frac{1+\cosx}{2}}$$ and therefore, he can write: $$\left|\cos\frac{x}{2}\right|=\sqrt{\frac{1+\frac59}{2}}$$ (4) Finally, from above equality, Cecil expressed $\cos\frac{x}{2}$ as: $$\cos\frac{x}{2}=\frac{\sqrt7}{3}$$
The mistake is in step (1). The trigonometric identity that Cecil used is not true. The correct equality should be: $$\cos^2x=1+\frac{56}{81}$$
The mistake is in step (2). Since $x\in\left(\frac32\pi,2\pi\right)$, it follows that $\cosx<0$. Therefore, it should be: $$\cosx=-\frac59$$
The mistake is in step (3). Cecil used the trigonometric identity incorrectly. It should be: $$\left|\cos\frac{x}{2}\right|=\sqrt{\frac{1-\frac{5}{9}}{2}}$$
The mistake is in step (4). Since $x\in\left(\frac32 \pi,2\pi\right)$ it follows that $\frac{x}{2}\in\left(\frac34\pi,\pi\right)$. Therefore, $\cos\frac{x}{2}<0$ and thus: $$\cos\frac{x}{2}=-\frac{\sqrt{7}}{3}$$