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The value of $\theta$ lying between $- \,\frac{\pi }{4}$ and $\frac{\pi }{2}$ and $0 \le A \le \frac{\pi }{2}$ and satisfying the equation $\left| {\begin{array}{*{20}{c}}{1\, + \,{{\sin }^2}A}&{{{\cos }^2}A}&{2\,\sin \,4\theta }\\{{{\sin }^2}A}&{1\, + \,{{\cos }^2}A}&{2\,\sin \,4\theta }\\{{{\sin }^2}A}&{{{\cos }^2}A}&{1\, + \,2\,\sin \,4\theta }\end{array}} \right|$ $= 0$ are :
$A =$ $\frac{\pi }{4}$ , $\theta =$ $- \,\frac{\pi }{8}$
$A =$ $\frac{{3\,\pi }}{8}$ $= \theta$
$A =$$\frac{\pi }{5}$ , $\theta =$ $- \,\frac{\pi }{8}$
All of the above
Solution
Use $R_1 \rightarrow R_1 -R_2$ and $R_2 \rightarrow R_2 -R_3$ and expand to get $D = 2 (1 + \sin \,4 \theta ) = 0 $
==>$\theta =$ $\frac{{n\,\pi }}{4}$ $- (- 1)^n$ $\frac{\pi }{8}$
==> independent of $A ==> A, B, C, $
