I need to prove trigonometric equation \tan 2x=\frac{2 \sin x \cdot

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

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.

Use a protractor to measure ${\displaystyle \angle a,\angle b}$, and ${\displaystyle \angle c}$. List the angle measures.

Prove the trigonometry identitied
${\displaystyle \frac{\sec A}{\sec A-1}-\frac{\sec A+1}{{\tan }^{2}A}=1}$

Given a triangles. One angle is ${\displaystyle {42}^{\circ }}$. Find the other angles if they're equal.

Prove this inequality with trigonometry $9{\cos }^{2}x-10\cos x\sin y-8\cos y\sin x+17\ge 1$

${\displaystyle \sin u=-\frac{24}{25}}$ (${\displaystyle u\in }$ Quadrant III)
${\displaystyle \cos v=-\frac{8}{17}}$ (${\displaystyle v\in }$ Quadrant II)
Evaluate the trigonometric function. Find ${\displaystyle \cos u,\sin v,\text{and}\sin (u-v)}$.