Prove the identity cotθ+cosecθ-1cotθ-cosecθ+1=1+cosθsinθ.
Proving cotθ+cosecθ-1cotθ-cosecθ+1=1+cosθsinθ:
Rewrite the left side using trigonometric identities,
cotθ+cosecθ-1cotθ-cosecθ+1=cotθ+cosecθ-(cosec2θ-cot2θ)cotθ-cosecθ+1
By perfect square formula,
cotθ+cosecθ-1cotθ-cosecθ+1=(cosecθ+cotθ)-cosecθ+cotθ(cosecθ-cotθ(1-cosecθ+cotθ)
Simplify the expression,
cotθ+cosecθ-1cotθ-cosecθ+1=(cosecθ+cotθ)(1-cosecθ+cotθ)(1-cosecθ+cotθ)
cotθ+cosecθ-1cotθ-cosecθ+1=cosecθ+cotθ
cotθ+cosecθ-1cotθ-cosecθ+1=1sinθ+cosθsinθ
cotθ+cosecθ-1cotθ-cosecθ+1=1+cosθsinθ
Hence, it is proved cotθ+cosecθ-1cotθ-cosecθ+1=1+cosθsinθ.