Prove the following. (secθ + tanθ) (1 - sinθ) = cosθ
Chapter 6 – Trigonometry – Text Book Solution
Problem set 6| Q 5.2| Page 138
Prove the following.
(2) (secθ + tanθ) (1 – sinθ) = cosθ
Solution
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.
Chapter 6 – Trigonometry – Text Book Solution
Problem Set 6 |Q 5.2| P 138
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