A) 1
B) \[sin\text{ }a\text{ }cos\text{ }a\]
C) 0
D) \[sin\text{ }x\text{ }cos\text{ }x\]
E) \[cosec\text{ }2x\]
Correct Answer: C
Solution :
\[\left| \begin{matrix} \cos (x-a) & \cos (x+a) & \cos x \\ \sin (x+a) & \sin (x-a) & \sin x \\ \cos a\tan x & \cos a\cos x & \cos ec2x \\ \end{matrix} \right|\] \[=\left| \begin{matrix} \cos (x-a)+\cos (x+a) & \cos (x+a) \\ \sin (x+a)+\sin (x-a) & \sin (x-a) \\ \cos a(\tan x+\cot x) & \cos a\,\cot x \\ \end{matrix} \right.\]\[\left. \begin{matrix} \cos x \\ \sin x \\ \cos ec2x \\ \end{matrix} \right|\] \[=\left| \begin{matrix} 2\cos x\cos a & \cos (x+a) & \cos x \\ 2\sin x\cos a & \sin (x-a) & \sin x \\ \cos a\left( \frac{{{\tan }^{2}}x+1}{\tan x} \right) & \cos a\cot x & \cos ec2x \\ \end{matrix} \right|\] \[=2\cos a\left| \begin{matrix} \cos x & \cos (x+a) & \cos x \\ \sin x & \sin (x-a) & \sin x \\ \frac{1+{{\tan }^{2}}x}{2\tan x} & \cos a\cot x & \cos ec2x \\ \end{matrix} \right|\] \[=2\cos a\left| \begin{matrix} \cos x & \cos (x+a) & \cos x \\ \sin x & \sin (x-a) & \sin x \\ \cos ec2x & \cos a\,\cot x & \cos ec2x \\ \end{matrix} \right|\] \[=2\cos a\times 0=0\]
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