Solution:
Given:
$\cos^2\text{x}+\sin\text{x}+1=0$
$\Rightarrow(1-\sin^2\text{x})+\sin\text{x}+1=0$
$\Rightarrow1-\sin^2\text{x}+\sin\text{x}+1=0$
$\Rightarrow\sin^2\text{x}-\sin\text{x}-2=0$
$\Rightarrow\sin^2\text{x}-2\sin\text{x}+\sin\text{x}-2=0$
$\Rightarrow\sin\text{x}(\sin\text{x}-2)+1(\sin\text{x}-2)=0$
$\Rightarrow(\sin\text{x}-2)(\sin\text{x}+1)=0$
$\Rightarrow\sin\text{x}-2=0$ or $\sin\text{x}+1=0$
$\Rightarrow\sin\text{x}=2$ or $\sin\text{x}=-1$
$\sin\text{x}=2$ is not possible.
$\Rightarrow\sin\text{x}=-1$
$\therefore\sin\text{x}=\sin\frac{3\pi}{2}$
$\Rightarrow\text{x}=\text{n}\pi+(-1)^\text{n}\frac{3\pi}{2},\ \text{n}\in\text{Z}$
The values of x lies in the third and fourth quadrants.
Hence, x lies in $\Big(\frac{5\pi}{4},\ \frac{7\pi}{4}\Big).$
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The number of terms with integral coefficients in the expansion of $\Big(17^{\frac{1}{3}}+35^{\frac{1}{2}}\text{x}\Big)^{600}$ is:
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