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\(\cfrac{sec\theta+tan\theta-1}{tan\theta-sec\theta+1}\)
\(=\cfrac{sec\theta+tan\theta-(sec^2\theta-tan^2\theta)}{tan\theta-sec\theta+1}\)
\(=\cfrac{sec\theta+tan\theta-(sec\theta+tan\theta)(sec\theta-tan\theta)}{tan\theta-sec\theta+1}\)
\(=\cfrac{(sec\theta+tan\theta)\{1-(sec\theta-tan\theta)\}}{tan\theta-sec\theta+1}\)
\(=\cfrac{(sec\theta+tan\theta)\cancel{(1-sec\theta+tan\theta)}}{\cancel{(1-sec\theta+tan\theta)}}\)
\(=sec\theta+tan\theta\)
\(=\cfrac{1}{cos\theta}+\cfrac{sin\theta}{cos\theta}\)
\(=\cfrac{1+sin\theta}{cos\theta}\) [প্রমানিত]