Question

proove

${\left(1-\tan x\right)}^2+{\left(1-\cot \right)}^2={\left(\sec x-\mathrm{cosecx}\right)}^2$

Answer 1

${\left(1-\tan x\right)}^2+{\left(1-\cot \right)}^2=1+{\tan }^{2}x-2\tan x+1+{\cot }^{2}x-2\cot x$
$=2+{\sec }^{2}x-1+{\mathrm{cosec}}^{2}x-1-2\left[\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}\right]$
\$={\sec }^{2}x+{\mathrm{cosec}}^{2}x-2\left[\frac{{\sin }^{2}x+{\cos }^{2}x}{\sin x\cos x}\right]$
$={\sec }^{2}x+{\mathrm{cosec}}^{2}x-2\left[\frac{1}{\sin x\cos x}\right]$
$={\sec }^{2}x+{\mathrm{cosec}}^{2}x-2\sec x\mathrm{cosecx}$
$={\left(\sec x-\cos ecx\right)}^2$