Question

sintheeta + 1- costheeta ÷ costheeta- 1+ sintheeta= 1+ sintheeta÷ costheeta

Answer 1

$LHS=\frac{sin\theta+1-cos\theta}{cos\theta-1+sin\theta}\\\\=\frac{\frac{sin\theta}{cos\theta}-\frac{1}{cos\theta}+\frac{cos\theta}{cos\theta}}{\frac{cos\theta}{cos\theta}-\frac{1}{cos\theta}+\frac{sin\theta}{cos\theta}}\\\\=\frac{tan\theta+sec\theta-1}{1-sec\theta+tan\theta}\\\\=\frac{tan\theta+sec\theta-(sec^2\theta-tan^2\theta)}{1-sec\theta+tan\theta}\\\\=\frac{tan\theta+sec\theta-(sec\theta+tan\theta)(sec\theta-tan\theta)}{1-sec\theta+tan\theta}\\\\=\frac{(sec\theta+tan\theta)(1-sec\theta+tan\theta)}{1-sec\theta+tan\theta}\\\\=sec\theta+tan\theta\\\\=\frac{1}{cos\theta}+\frac{sin\theta}{cos\theta}\\\\=\frac{1+sin\theta}{cos\theta}=RHS$