Question

cos theta minus sin theta plus one divided by cos theta plus sin theta minus one

Answer 1

$\large\sf\underline{Given}$

$\sf\:\frac{Cosθ-Sinθ+1}{Cosθ+Sinθ-1}$

$\large\sf\underline{Solution}$

Dividing the numerator and denominator by Sin θ

$\large\sf⇨\:\frac{\frac{Cosθ}{Sinθ}-\frac{Sinθ}{Sinθ}+\frac{1}{Sinθ}}{\frac{Cosθ}{Sinθ}+\frac{Sinθ}{Sinθ}-\frac{1}{Sinθ}}$

$\sf⇨\:\frac{Cosθ-1+Cosecθ}{Cotθ+1-Cosecθ}$

$\sf⇨\:\frac{Cotθ+Cosecθ-1}{Cotθ-Cosecθ+1}$

$\sf⇨\:\frac{Cotθ+Cosecθ-(Cosec^{2}θ-Cot^{2} θ)}{Cotθ-Cosecθ+1}$

$\tt\pink{[∵\:Cosec^{2}θ=1+Cot^{2}]}$

$\sf⇨\:\frac{Cotθ+Cosecθ-[(Cosecθ-Cotθ)(Cosecθ+Cotθ)]}{Cotθ-Cosecθ+1}$

$\sf⇨\:\frac{Cotθ+Cosecθ[1-Cosecθ+cotθ]}{1-Cosecθ+cotθ}$

$\sf⇨\:Cotθ+cosecθ$

• Hope uh understand :)