Question

If $x+y={45}^{\circ }$, then prove that :
(a) $\left(1+\tan x\right)\left(1+\tan y\right)=2$

(b) $\left(\cot x-1\right)\left(\cot y-1\right)=2$

Answer 1

$x+y=45\left(a\right)\tan 45=\tan (x+y)1=\frac{\tan x+\tan y}{1-\tan x\tan y}1+1-\tan x\tan y=\tan x+\tan y+12=(1+\tan x)(1+\tan y)\left(b\right)\cot 45=\cot (x+y)1=\frac{\cot x\cot y+1}{\cot y-\cot x}+1+\cot y-\cot x-\cot x\cot y=1+1(\cot x-1)(\cot y-1)=2$