A) \[{{\Delta }_{1}}-{{\Delta }_{2}}=x(cos2\theta -cos4\theta )\]
B) \[{{\Delta }_{1}}+{{\Delta }_{2}}=-2{{x}^{3}}\]
C) \[{{\Delta }_{1}}-{{\Delta }_{2}}=-2{{x}^{3}}\]
D) \[{{\Delta }_{1}}+{{\Delta }_{2}}=-2({{x}^{3}}+x-1)\]
Correct Answer: B
Solution :
\[{{\Delta }_{1}}=f(\theta )=\left| \begin{matrix} x & \sin \theta & \cos \theta \\ -\sin \theta & -x & \text{l} \\ \cos \theta & \text{l} & x \\ \end{matrix} \right|=-{{x}^{3}}\] and\[{{\Delta }_{2}}=f(2\theta )=\left| \begin{matrix} x & \sin 2\theta & \cos 2\theta \\ -\sin 2\theta & -x & \text{l} \\ \cos 2\theta & \text{l} & x \\ \end{matrix} \right|=-{{x}^{3}}\] So\[{{\Delta }_{1}}+{{\Delta }_{2}}=-2{{x}^{3}}\]
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