Interval [0,2pi)2 cos^2 x - cos x = 0?

Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 11 passengers per minute.

If ${\displaystyle \sin x+\sin y=a\text{and}\cos x+\cos y=b}$ then find ${\displaystyle \tan (x-\frac{y}{2})}$

Verify that the equation is an identity. (Hint: $\cos 2x=\cos (x+x)$.)
$\cos 2x+2{\sin }^{2}x-1=0$

Equation in the complex plane $8z=i|z{|}^3\overline{z}$?

I calculated ${\displaystyle {\sin 75}^{\circ }}$ as ${\displaystyle \frac{1}{2\sqrt{2}}+\frac{\sqrt{3}}{2\sqrt{2}}}$, but the answer is ${\displaystyle \frac{\sqrt{2}+\sqrt{6}}{4}}$. What went wrong?
I calculated the exact value of ${\displaystyle {\sin 75}^{\circ }}$ as follows:
${\displaystyle {\sin 75}^{\circ }=\sin ({30}^{\circ }+{45}^{\circ })}$
$=\sin {30}^{\circ }\cos {45}^{\circ }+\cos {30}^{\circ }\sin {45}^{\circ }$
$=\frac{1}{2}\cdot \frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}\cdot \frac{1}{\sqrt{2}}$
$=\frac{1}{2\sqrt{2}}+\frac{\sqrt{3}}{2\sqrt{2}}$
The actual answer is
${\displaystyle \frac{\sqrt{2}+\sqrt{6}}{4}}$

Given tanx is ${\displaystyle \frac{3}{4}}$. you can use the double formula to find ${\displaystyle \tan 2x}$. the teacher wants us to find 3 other doubles like ${\displaystyle \sin 2x}$ or ${\displaystyle \cos 2x}$ without using the double angle formulas.

Verify the identity ${\displaystyle \sec \theta \cot \theta -\sin \theta =\frac{{\cos }^2\theta }{\sin \theta }}$