Simplify (1+cotθ+tanθ)(sinθ−cosθ)sec3θ−cosec3θ
Answer:
sin2θ cos2θ
- (1+cotθ+tanθ)(sinθ−cosθ)sec3θ−cosec3θ=(1+cosθsinθ+sinθcosθ)(sinθ−cosθ)(secθ−cosecθ)(sec2θ+secθ×cosecθ+cosec2θ) [Since, a3−b3=(a−b)(a2+ab+b2]=(sinθ cosθ+cos2θ+sin2θsinθ cosθ)(sinθ−cosθ)(1cosθ−1sinθ)(1cos2θ+1sinθ cosθ+1cos2θ) [Since, secθ=1cosθ and cosecθ=1sinθ]=(1+sinθ cosθsinθ cosθ)(sinθ−cosθ)(sinθ−cosθ)(sin2θ+sinθ cosθ+cos2θ)(sinθ cosθ)(sin2θ cos2θ)=(1+sinθ cosθ)(sinθ−cosθ)(sin3θ cos3θ)(sinθ cosθ)(sinθ−cosθ)(1+sinθ cosθ)=sin2θ cos2θ