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

Starting with the left-hand side (LHS):

(secθ + tanθ)(1 – sinθ)

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

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

= secθ – sinθ secθ + sinθ/cosθ – sinθ sinθ/cosθ // using the identity tanθ = sinθ/cosθ

= secθ – sinθ secθ + sinθ/cosθ – sin²θ/cosθ // combining like terms

= secθ – sinθ secθ + cos²θ sinθ/cosθ – sin²θ/cosθ // using the identity sin²θ + cos²θ = 1 and multiplying the numerator and denominator of the second term by cosθ/cosθ

= secθ – sinθ secθ + sinθ cosθ – sin²θ/cosθ // simplifying

= secθ – sinθ secθ + sinθ cosθ – sinθ cosθ/cosθ // multiplying the numerator and denominator of the third term by cosθ/cosθ

= secθ – sinθ secθ + sinθ cosθ – sinθ // simplifying

= secθ – sinθ(1 + secθ – 1) + sinθ cosθ // adding and subtracting 1

= secθ – sinθ + sinθ cosθ – sinθ secθ // rearranging terms

= cosθ + sinθ cosθ – sinθ secθ // using the identity cosθ = 1/secθ

= cosθ(1 – sinθ + tanθ) // factorizing out cosθ

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

(secθ + tanθ)(1 – sinθ) = cosθ.