\((1)\)求值：\({co}{{{s}}^{4}}{{15}^{0}}-{si}{{{n}}^{4}}{{15}^{0}}= \)________．

\((2)\)已知\({|}a{|=}1\)，\({|}b{|=}2\)，若\(a⊥(a+b)\)，则向量\(a\)与\(b\)的夹角为__________．

\((3)\)数列\(\{a_{n}\}\)满足\(a_{1}=0\)，\(a_{n+1}=\dfrac{{{a}_{n}}-\sqrt{3}}{\sqrt{3}{{a}_{n}}+1}\) \((n∈N^{*})\)，则\(a_{2015}=\)________．

\((4)\)已知数列\(\left\{ {{a}_{n}} \right\}\)是各项均不为零的等差数列，\({{S}_{n}}\)为其前\(n\)项和，且\({{a}_{n}}=\sqrt{{{S}_{2n-1}}}\left( n\in {{N}^{*}} \right).\)若不等式\(\dfrac{\lambda }{{{a}_{n}}}\leqslant \dfrac{n+8}{n}\)对任意\(n\in {{N}^{*}}\)恒成立，则实数\(\lambda \)的最大值为_____________．

设平面内的向量\( \overset{→}{OA}=(-1,-3), \overset{→}{OB}=(5,3), \overset{→}{OM}=(2,2) \)，点\(P\)在直线\(OM\)上，且\( \overset{→}{PA}· \overset{→}{PB}=-16 \)．
\((1)\)求\( \overset{→}{OP} \)的坐标；
\((2)\)求\(∠APB\)的余弦值；
\((3)\)设 \(t\)\(∈R\)，求\(| \overset{→}{OA}+t \overset{→}{OP}| \)的最小值．

过圆\({{x}^{2}}+{{y}^{2}}-6x-8y+21=0\)上一动点\(P\)作圆\({{x}^{2}}+{{y}^{2}}=4\)的两条切线，切点分别为\(A,B\)，设向量\(\overrightarrow{PA},\overrightarrow{PB}\)的夹角为\(\theta \)，则\(\cos \theta \)的取值范围为          ．

\((1)\) 直线\(x+2ay-1=0\)与直线\((a-1)x-ay-1=0\)平行，则\(a\)的值是_________．

\((2)\)　在面积为\(S\)的\(\triangle ABC\)的边\(AB\)上任取一点\(P\)，则\(\triangle PBC\)的面积不小于\( \dfrac{S}{3}\)的概率是_________

\((3)\)已知直线\(l\)：\(x- \sqrt{3}y+6=0 \)与圆\(x^{2}+y^{2}=12\)交于\(A\)，\(B\)两点，过\(A\)，\(B\)分别作\(l\)的垂线与\(x\)轴交于\(C\)，\(D\)两点，则\(\left|CD\right|= \)_____________．

\((4)\)在平面直角坐标系\(xoy\)中，直线\(y=-x+2\)与圆\({{x}^{2}}+{{y}^{2}}={{r}^{2}}(r > 0)\)交于\(A\)，\(B\)两点，\(O\)为坐标原点，若圆上有一个\(C\)满\(\overset{\to }{{OC}}\,=\dfrac{5}{4}\overset{\to }{{OA}}\,+\dfrac{3}{4}\overset{\to }{{OB}}\,\)，则\(r=\)______________．

已知函数\(f(x)=\dfrac{1}{x+1}\)，点\(O\)为坐标原点，点\({{A}_{n}}(n{ },f(n))(n\in {{N}^{*}})\)，向量\(\overrightarrow{i}=(0,1)\)，\({{\theta }_{n}}\)是向量\(\overrightarrow{O{{A}_{n}}}\)与\(\overrightarrow{i}\)的夹角，则\( \dfrac{\cos {θ}_{1}}{\sin {θ}_{1}}+ \dfrac{\cos {θ}_{2}}{\sin {θ}_{2}}+...+ \dfrac{\cos {θ}_{2017}}{\sin {θ}_{2017}} \)的值为______．