若\(\sin 2\alpha =\dfrac{\sqrt{5}}{5}\)，\(\sin \left( \beta -\alpha \right)=\dfrac{\sqrt{10}}{10}\)，且\(α∈\left[ \dfrac{π}{4},π\right] \)，\(β∈\left[π, \dfrac{3}{2}π\right] \)，则\(\alpha +\beta \)的值\((\)   \()\)

\((1)\)已知角\(α\)终边上一点\(P(-4,3)\)，求\(\dfrac{\cos \left( \dfrac{π}{2}+α\right)\sin \left(-π-α\right)}{\cos \left( \dfrac{11π}{2}-α\right)\sin \left( \dfrac{9π}{2}-+α\right)} \)的值．

\((2)\)若\(\sin x=\dfrac{m-3}{m+5} \)，\(\cos x=\dfrac{4-2m}{m+5} \)，\(x∈(\dfrac{π}{2} ,π)\)，求\(\tan x\)．

已知\(\cos (\dfrac{\pi }{4}+x)=\dfrac{3}{5},\dfrac{17}{12}\pi < x < \dfrac{7}{4}\pi \)，则\(\dfrac{\sin 2x+2{{\sin }^{2}}x}{1-\tan x}\)的值为__________。

已知\(f\left( \alpha \right)=\dfrac{\cos \left( \dfrac{\pi }{2}+\alpha \right)\cdot \cos \left( 2\pi -\alpha \right)\cdot \sin \left( -\alpha +\dfrac{3}{2}\pi \right)}{\sin \left( -\pi -\alpha \right)\sin \left( \dfrac{3}{2}\pi +\alpha \right)}\) ．

\((1)\)化简\(f\left( \alpha \right)\)；\((2)\)若\(\alpha \)是第三象限角，且\(\cos \left( \alpha -\dfrac{3}{2}\pi \right)=\dfrac{1}{5}\)，求\(f\left( \alpha \right)\)的值．

已知\(\sin \alpha +2\cos \alpha =\dfrac{\sqrt{10}}{2}\)，则\(\tan 2α=\)

已知\(0 < α < \dfrac{\pi }{2}\)，\({-}\dfrac{\pi }{2} < β < 0\)，\(\cos \)\((α-β)={-}\dfrac{5}{13}\)，\(\sin \)\(α=\dfrac{4}{{5}}\)，则\(\sin \)\(β\)的值为 ______．

已知函数\(f(α)= \dfrac{\cos ( \dfrac{π}{2}+α)\cos (2π+α)\sin (-α+ \dfrac{3}{2}π)}{\sin (α+ \dfrac{7}{2}π)\sin (-3π-α)} \)．

\((1)\)化简\(f(α)\)；

\((2)\)若\(α\)是第三象限角，且\(\cos (α- \dfrac{3}{2} π)= \dfrac{1}{5} \)，求\(f(α)\)．