Question

If $x=a(\sec \theta +\tan \theta {)}^2,y=b(\sec \theta -\tan \theta {)}^2$, then $x^2{y}^2=$

• A. $ab\sec \theta$
• B. $a^2{b}^2\tan \theta$
• C. $a^4{b}^4$
• D. $a^2{b}^2$

Answer 1

The correct option is D $a^2{b}^2$

$x=a{(\sec \theta +\tan \theta )}^2$

$\Rightarrow y=b{(\sec \theta -\tan \theta )}^2$

$\Rightarrow x.y=ab\left[{(\sec \theta +\tan \theta )}^2{(\sec \theta -\tan \theta )}^2\right]$

$=ab{[{\sec }^2\theta -{\tan }^2\theta ]}^2$

$=ab{\left[1\right]}^2[\because 1+{\tan }^2\theta ={\sec }^2\theta ]$

$=ab$

$\Rightarrow x^2{y}^2=a^2{b}^2$

Hence, the answer is $a^2{b}^2.$