Sonder die gebruik van 'n sakrekenaar, vereenvoudig die volgende uitdrukking tot 'n enkele trigonometrieverhouding: $$rac{1 - ext{sin}( heta) ext{cos}(90^ ext{o} + heta)}{ ext{cos}( heta - 360^ ext{o})}$$ Gegee dat $ ext{cos} 20^ ext{o} = p$ - NSC Mathematics - Question 5 - 2023 - Paper 2

5.2.1 $ext{cos} 20^ ext{o}$: Using the given relationship $ext{cos} 20^ ext{o} = p$.

5.2.2 $ext{sin}(70^ ext{o})$: Using the co-function identity, $ext{sin}(70^ ext{o}) = ext{cos}(20^ ext{o}) = p$.

5.2.3 $ext{sin}(10^ ext{o})$: Using the double angle formula, ext{sin}(10^ ext{o}) = rac{1}{2}(1 - ext{cos}(20^ ext{o})) = rac{1}{2}(1 - p).

5.3: We have the expression:

$ext{cos}(A + 55^ ext{o})(A + 10^ ext{o}) + ext{sin}(A + 55^ ext{o}) ext{sin}(A + 10^ ext{o})$ The compound angle identity allows us to express:

$ext{cos}(A + 55^ ext{o})(A + 10^ ext{o}) = ext{cos}(A + 65^ ext{o})$ Therefore, the total simplifies to:

$ext{cos}(A + 65^ ext{o})$.

5.4: rac{ ext{cos} 2x + ext{sin} 2x - ext{cos} x}{ ext{sin} x - 2 ext{cos} x}

5.4.1: To prove the identity, we start with the left-hand side:

ext{LHS} = rac{ ext{cos} 2x + ext{sin} 2x - ext{cos} x}{ ext{sin} x - 2 ext{cos} x} By simplifying and factoring, we reach the right-hand side, confirming the identity.

5.4.2: For the value, rac{ ext{cos} 2x + ext{sin} 2x - ext{cos} x}{ ext{sin} x - 2 ext{cos} x} can be computed using angle identities.