Question

If $\sin \left(\theta \right)+\cos \left(\theta \right)=m$ and $\sec \left(\theta \right)+\mathrm{cosec}\left(\theta \right)=n$, 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

$2m$

Solve by substituting the values in given expression:

Put the given values $\sin \left(\theta \right)+\cos \left(\theta \right)=m$ and $\sec \left(\theta \right)+\mathrm{cosec}\left(\theta \right)=n$ in the required expression, then

$n\left(m+1\right)\left(m-1\right)=\left(\sec \left(\theta \right)+\mathrm{cosec}\left(\theta \right)\right)\left(\sin \left(\theta \right)+\cos \left(\theta \right)+1\right)\left(\sin \left(\theta \right)+\cos \left(\theta \right)-1\right)$

Apply the algebraic identity ${\mathit{a}}^{\mathbf{2}}\mathbf{-}{\mathit{b}}^{\mathbf{2}}\mathbf{=}\left(a+b\right)\left(a-b\right)$, then

$\begin{array}{rcl}n\left(m+1\right)\left(m-1\right)& =& \left(\sec \left(\theta \right)+\mathrm{cosec}\left(\theta \right)\right)\left({\left(\sin \left(\theta \right)+\cos \left(\theta \right)\right)}^2-1^2\right)\end{array}$

Now, applying the algebraic identity ${\left(a+b\right)}^{\mathbf{2}}\mathbf{=}{\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{b}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{a}\mathit{b}$, we get

$\begin{array}{rcl}\therefore n+\left(m+1\right)\left(m-1\right)& =& \left(\sec \left(\theta \right)+\mathrm{cosec}\left(\theta \right)\right)\left({\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)+2\sin \left(\theta \right)\cos \left(\theta \right)-1\right)\\ & =& \left(\sec \left(\theta \right)+\mathrm{cosec}\left(\theta \right)\right)\left(1+2\sin \left(\theta \right)\cos \left(\theta \right)-1\right)\\ & =& \left(\frac{1}{\cos \left(\theta \right)}+\frac{1}{\sin \left(\theta \right)}\right)\left(2\sin \left(\theta \right)\cos \left(\theta \right)\right)\\ & =& \frac{2\sin \left(\theta \right)\cos \left(\theta \right)}{\cos \left(\theta \right)}+\frac{2\sin \left(\theta \right)\cos \left(\theta \right)}{\sin \left(\theta \right)}\\ & =& 2\left(\sin \left(\theta \right)+\cos \left(\theta \right)\right)\\ & =& 2m\end{array}$

Hence, the correct option is (C).