$\Rightarrow$ $\frac{1}{{{{\sin }^2}\theta \,{{\cos }^2}\theta }}\,\, = \,8$
$sin^22\theta = 1/2 = {\left( {\frac{1}{{\sqrt 2 }}} \right)^2}$
$2\theta = n\pi + \pi /4$
$\theta = n\pi /2 + \pi /8$
for $n = 0$ $\Rightarrow$ $\theta = \pi /8$
for $n=1$ $\Rightarrow$ $\theta = 3\pi /8$
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$f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x<1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x<2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.$
where a,b,c $\in R$ and $[t]$ denotes greatest integer less than or equal to $t.$ Then, which of the following statements is true $?$