Question

The value of $\left|\begin{array}{ccc}\cos (x-a)& \cos (x+a)& \cos x\\ \sin (x+a)& \sin (x-a)& \sin x\\ \cos a\tan x& \cos a\cot x& \cos ec2x\end{array}\right|$ is equal to:

• A. 1
• B. $sinacosa$
• C. 0
• D. $sinxcosx$
• E. $cosec2x$

Answer 1

Answer: (C)

$\left|\begin{array}{ccc}\cos (x-a)& \cos (x+a)& \cos x\\ \sin (x+a)& \sin (x-a)& \sin x\\ \cos a\tan x& \cos a\cos x& \cos ec2x\end{array}\right|$ $=|\begin{array}{cc}\cos (x-a)+\cos (x+a)& \cos (x+a)\\ \sin (x+a)+\sin (x-a)& \sin (x-a)\\ \cos a(\tan x+\cot x)& \cos a\cot x\end{array}$ $\begin{array}{c}\cos x\\ \sin x\\ \cos ec2x\end{array}|$ $=\left|\begin{array}{ccc}2\cos x\cos a& \cos (x+a)& \cos x\\ 2\sin x\cos a& \sin (x-a)& \sin x\\ \cos a\left(\frac{{\tan }^{2}x+1}{\tan x}\right)& \cos a\cot x& \cos ec2x\end{array}\right|$ $=2\cos a\left|\begin{array}{ccc}\cos x& \cos (x+a)& \cos x\\ \sin x& \sin (x-a)& \sin x\\ \frac{1+{\tan }^{2}x}{2\tan x}& \cos a\cot x& \cos ec2x\end{array}\right|$ $=2\cos a\left|\begin{array}{ccc}\cos x& \cos (x+a)& \cos x\\ \sin x& \sin (x-a)& \sin x\\ \cos ec2x& \cos a\cot x& \cos ec2x\end{array}\right|$ $=2\cos a\times 0=0$