Question

If $(\sec A+\tan A-1)(\sec A-\tan A+1)=m\tan A$.Find $m$.

Answer 1

$(\sec A+(\tan A-1))(\sec A-(\tan A-1))=m\tan A$

$R.H.S.-$

$=m\tan A$

$L.H.S.-$

$(\sec A+(\tan A-1))(\sec A-(\tan A-1)){\sec }^{2}A-{(\tan A-1)}^2$ .......... $a^2-b^2=(a-b)(a+b)$

$\Rightarrow {\sec }^{2}A-({\tan }^{2}A+1-2\tan A)$ ............. $(a-b{)}^2=a^2-2ab+b^2$

$\Rightarrow {\sec }^{2}A-{\tan }^{2}A-1+2\tan A$

$\Rightarrow 1-1+2\tan A............(\because 1+{\tan }^{2}A={\sec }^{2}A)$

$\Rightarrow 2\tan A$

On comparing the values of L.H.S. and R.H.S., we have

$m=2$

Hence, the correct answer is 2.