Prove each of the following identities:

Consider L.H.S. = (1 + tan θ + cot θ)(sin θ – cos θ)

= sin θ – cos θ + tan θ sin θ – tan θ cos θ + cot θ sin θ – cot θ cos θ

= sin θ – cos θ + tan θ sin θ – sin θ + cos θ – cot θ cos θ

= tan θ sin θ – cot θ cos θ

= (sin² θ/cos θ) – (cos² θ/sin θ)

= [(1/cos θ) × (1/cosec² θ)] – [(1/cosec θ) × (1/sec² θ)]

= (sec θ/cosec² θ) – (cosec θ/sec² θ)

= R.H.S.

Hence, proved.