Question

If $\sin \theta +\cos \theta =mand\sec \theta +\cos ec\theta =n,$ then $n\left(m+1\right)\left(m-1\right)=$

• A. $m$
• B. $n$
• C. $2m$
• D. $2n$

Answer 1

The correct option is C $2m$

$n\left(m+1\right)\left(m-1\right)$

$=\left(\sec \theta +\cos ec\theta \right)\left(\sin \theta +\cos \theta +1\right)\left(\sin \theta +\cos \theta -1\right)$

$=\left(\sec \theta +\cos ec\theta \right)\left[{\left(\sin \theta +\cos \theta \right)}^2-1^2\right]$

$=\left(sec\theta +\cos ec\theta \right)\left({\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta -1\right)$

$=\left(\sec \theta +\cos ec\theta \right)\left(2\sin \theta \cos \theta \right)$

$=2\sin \theta +2\cos \theta$

$=2m$