Trigonometry Problems

$(1).\,\,\,$ Evaluate $\dfrac{\cos{A}-\cos{3A}}{\cos{A}}$ $+$ $\dfrac{\sin{A}+\sin{3A}}{\sin{A}}$

$(2).\,\,\,$ Find the value of $\tan(\theta +\alpha) \cot \alpha$ if $\sin \theta = n \sin(\theta + 2\alpha)$

$(3).\,\,\,$ Simplify $\dfrac{\sin(n+1)\alpha -\sin(n-1)\alpha}{\cos(n+1)\alpha +2\cos n\alpha +\cos(n-1)\alpha}$

$(4).\,\,\,$ Find $a \cos 2\alpha$ $+$ $b\sin 2\alpha$ if $\dfrac{\cos \alpha}{a} = \dfrac{\sin \alpha}{b}$

$(5).\,\,\,$ Simplify the trigonometric expression $\sin x \tan x + \cos x$

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