Question

Sin theta + cos theta =m ,tan theta +cot theta =n eliminate theta in each of the following

Answer 1

$\textbf{Given:}$

$m=sin\theta+cos\theta$

$n=tan\theta+cot\theta$

$\textbf{To find:}$

$\text{Eliminating \theta from the given equations}$

$\textbf{Solution:}$

$\text{Consider,}$

$m=sin\theta+cos\theta$

$\text{Squaring on bothsies, we get}$

$m^2=(sin\theta+cos\theta)^2$

$m^2=sin^2\theta+cos^2\theta+2\,sin\theta\,cos\theta$

$m^2=1+2\,sin\theta\,cos\theta$.........(1)

$\text{and}$

$n=tan\theta+cot\theta$

$n=\dfrac{sin\theta}{cos\theta}+\dfrac{cos\theta}{sin\theta}$

$n=\dfrac{sin^2\theta+cos^2\theta}{cos\theta\;sin\theta}$

$n=\dfrac{1}{cos\theta\;sin\theta}$

$cos\theta\;sin\theta=\dfrac{1}{n}$........(2)

$\text{Using (2) in (1), we get}$

$m^2=1+2(\dfrac{1}{n})$

$m^2=1+\dfrac{2}{n}$

$m^2=\dfrac{n+2}{n}$

$m^2n=n+2$

$\implies\bf\,m^2n-n-2=0$

$\textbf{Answer:}$

$\textbf{The required relation is}$

$\boxed{\bf\,m^2n-n-2=0}$