Prove the following. (1) secθ(1 - sinθ) (secθ + tanθ) = 1

Starting with the left-hand side (LHS):

secθ(1 – sinθ)(secθ + tanθ)

= secθ(1 – sinθ)secθ + secθ(1 – sinθ)tanθ // distributive property

= sec²θ – sinθ secθ + secθ tanθ – sinθ secθ tanθ // multiplying out

= sec²θ + secθ tanθ – sinθ secθ (1 + tan²θ) // factoring

= sec²θ + secθ tanθ – sinθ secθ sec²θ // using the identity 1 + tan²θ = sec²θ

= sec²θ + secθ tanθ – sec²θ sinθ = sec²θ (1 – sinθ + tanθ) // factorizing out sec²θ

= 1 / cos²θ (cos²θ) // using the identity 1/cosθ = secθ

= 1 // simplifying

Therefore, the left-hand side (LHS) is equal to the right-hand side (RHS), so we have proved that:

secθ(1 – sinθ)(secθ + tanθ) = 1.