Answer
See below.
Work Step by Step
We need to show $1+cos\theta+sin\theta=(1+cos\theta-sin\theta)(sec\theta+tan\theta)$.
We have $RHS=sec\theta+tan\theta+cos\theta sec\theta+cos\theta tan\theta-sin\theta sec\theta-sin\theta tan\theta=sec\theta+tan\theta+1+sin\theta-tan\theta-sin\theta tan\theta=1+sec\theta+sin\theta-sin\theta tan\theta=1+sin\theta+\frac{1-sin^2\theta}{cos\theta}=1+sin\theta+cos\theta=LHS$