Question

Value of the determinant
$\left|\begin{array}{ccc}\sec x& \sin x& \tan x\\ 0& 1& 0\\ \tan x& \cot x& \sec x\end{array}\right|$ is given by___

Also try to think on the lines that expanding along which row or column will make the calculation easier.

Answer 1

Here we can see that second row has two entries as 0. So it’s better if we expand the determinant with respect to the second row. So doing it with respect to second row it comes out like this.

$(-1{)}^{2+1}\times 0\times (\sin x\sec x-\cot x\tan x)$
$+1\times (-1{)}^{2+2}\times (\sec x\sec x-\tan x\tan x)$
$+0\times (-1{)}^{2+3}\times (\sec x\cot x-\sin x\tan x)$
$=0+1\times 1\times ({\sec }^{2}x-{\tan }^{2}x)+0\text{Since}{\sec }^{2}x-{\tan }^{2}x=1.\text{So determinant = 0 + 1 + 0}=1$