Question

If $(\sec \theta +\tan \theta )(\sec \varphi +\tan \varphi )(\sec \psi +\tan \psi )=\tan \theta \tan \varphi \tan \psi$ ,then $(\sec \theta -\tan \theta )(\sec \varphi -\tan \varphi )(\sec \psi -\tan \psi )$ is equal to

• A. $\cot \theta \cot \varphi \cot \psi$
• B. $\tan \theta \tan \varphi \tan \psi$
• C. $\tan \theta +\tan \varphi +\tan \psi$
• D. $\cot \theta +\cot \varphi +\cot \psi$

Answer 1

The correct option is A $\cot \theta \cot \varphi \cot \psi$

$1+{\tan }^{2}x={\sec }^{2}x\Rightarrow 1={\sec }^{2}x-{\tan }^{2}x\Rightarrow 1=(\sec x+\tan x)(\sec x-\tan x)$

$\Rightarrow (\sec x-\tan x)=\frac{1}{(\sec x+\tan x)}$

Therefore
$(\sec \theta -\tan \theta )(\sec \varphi -\tan \varphi )(\sec \psi -\tan \psi )$

$=\frac{1}{(\sec \theta +\tan \theta )(\sec \varphi +\tan \varphi )(\sec \psi +\tan \psi )}$

$=\frac{1}{\tan \theta \tan \varphi \tan \psi }=\cot \theta \cot \varphi \cot \psi$