Question

If $cosecA+\cot A=m$, show that $\frac{m^2-1}{m^2+1}=\cos A$.

Answer 1

Let, $cosecA+\cot A=m$

Then

$\frac{m^2+1}{m^2-1}$

$=\frac{cose{c}^{2}A+{\cot }^{2}A+2cosecA.\cot A+1}{cose{c}^{2}A+{\cot }^{2}A+2cosecA.\cot A-1}$

$=\frac{2cose{c}^{2}A+2cosecA.\cot A}{2{\cot }^{2}A+2cosecA.\cot A}$ [Since $cose{c}^{2}A-1={\cot }^{2}A$ and $1+{\cot }^{2}A=cose{c}^{2}A$]

$=\frac{cosecA}{\cot A}=\sec A$.........(1).

Now

$\frac{m^2-1}{m^2+1}=\cos A$ [Using (1)]