Prove the following.
(1) secθ(1 – sinθ) (secθ + tanθ) = 1
Chapter 6 – Trigonometry – Text Book Solution
Problem set 6| Q 5.1| Page 138
Prove the following.
(1) secθ(1 – sinθ) (secθ + tanθ) = 1
Solution
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.
Chapter 6 – Trigonometry – Text Book Solution
Problem Set 6 |Q 5.1| P 138
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