Trigonometry problems

$(1).\,\,\,$ If $A+B = 45^°$, then find $(1+\tan{A})(1+\tan{B})$

$(2).\,\,\,$ Simplify $\dfrac{\cos{(90^°-\theta)}}{1+\sin{(90^°-\theta)}}$ $+$ $\dfrac{1+\sin{(90^°-\theta)}}{\cos{(90^°-\theta)}}$

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

$(4).\,\,\,$ Find the value of $\tan(\theta +\alpha) \cot \alpha$ if $\sin \theta = n \sin(\theta + 2\alpha)$

$(5).\,\,\,$ Simplify $\dfrac{\sin(n+1)\alpha -\sin(n-1)\alpha}{\cos(n+1)\alpha +2\cos n\alpha +\cos(n-1)\alpha}$

$(6).\,\,\,$ Find $a \cos 2\alpha$ $+$ $b\sin 2\alpha$ if $\dfrac{\cos \alpha}{a} = \dfrac{\sin \alpha}{b}$

$(7).\,\,\,$ Simplify the trigonometric expression $\sin x \tan x + \cos x$

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