Prove the following identities

secθ (1 – sinθ)(secθ + tanθ) = 1

Consider LHS,

LHS = secθ (1 – sinθ) (secθ + tanθ)

⇒ secθ (1 – sinθ) (secθ + tanθ) = (secθ - ) (secθ + tanθ)

We know that = tanθ.

⇒ secθ (1 – sinθ) (secθ + tanθ) = (secθ – tanθ) (secθ + tanθ)

We know that (a + b) (a – b) = a² – b².

⇒ secθ (1 – sinθ) (secθ + tanθ) = sec²θ – tan²θ

We know that sec²θ – tan²θ = 1.

∴ secθ (1 – sinθ) (secθ + tanθ) = 1 = RHS

Hence proved.