Solve sec(30)+tan(45)-csc(60)/sec(30)+cos(60)+cot(45)

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 ${\displaystyle \tan (x-\frac{y}{2})}$

If $\alpha$, $\beta$, $\gamma$, $\delta$ are smallest positive angle in ascending order of magnitude which have their sines equal to positive quantity $\lambda$ , then if
$4\sin \frac{\alpha }{2}+3\sin \frac{\beta }{2}+2\sin \frac{\gamma }{2}+\sin \frac{\delta }{2}=2\sqrt{1+\lambda }$
then $\lambda$ = ?
Options-
(1)$\sin \alpha$
(2)$\sin \beta$
(3)$\sin \gamma$
(4)$\sin \delta$

Solve the equation $\frac{3a{\sin }^2\left(\theta \right)\cos \left(\theta \right)}{3a{\cos }^2\left(\theta \right)\sin \left(\theta \right)}=\tan \left(\theta \right)$

Prove that
${\displaystyle \left(\tan x\right)\left(\sin 2x\right)=2–2{\cos }^2}$

Solve the equation $3+\sin \theta =\frac{6-\sqrt{2}}{2}$

If ${\displaystyle \cos x=-\frac{12}{13}\text{and}\csc x<0}$,
find ${\displaystyle \cot \left(2x\right)}$