Gegee: tan(π + A) · cos(180° - A) - sin(360° - A) sin(2π + A) 4.1.1 Vereenvoudig deur reduksie: tan(π + A) 4.1.2 Vereenvoudig: tan(π + A) · cos(180° - A) - sin(360° - A) sin(2π + A) 4.2 Voltooi die identiteit: cot² x - cosec² x = 4.3 Bewys die identiteit: sin x + cos² x - cosec x = cosec x - NSC Technical Mathematics - Question 4 - 2024 - Paper 2

Using angle identities, we can simplify both terms:

1. For $cos(180° - A)$, it equals $-cos A$;
2. For $sin(360° - A)$, it equals $-sin A$;
3. For $sin(2π + A)$, it equals $sin A$,

Substituting these into our equation:

rac{ an(π + A) imes (- ext{cos} A) + ext{sin} A}{ ext{sin} A}

Thus, we have:

= rac{ an A imes - ext{cos} A + ext{sin} A}{ ext{sin} A}

This further simplifies to:

an A imes -rac{ ext{cos} A}{ ext{sin} A} + 1

Which concludes that:

$= - ext{cot A} + 1$

We start with the left-hand side:

1. Recall the identity: $ext{cot}^2 x + 1 = ext{cosec}^2 x$
2. Rearranging gives:

$ext{cot}^2 x = ext{cosec}^2 x - 1$

So we can conclude that:

$cot² x - cosec² x = -1$

Starting from the left-hand side:

1. Recognize that ext{cosec} x = rac{1}{ ext{sin} x};
2. This gives us:

ext{LHS} = ext{sin} x + ext{cos}^2 x - rac{1}{ ext{sin} x} 3. Now we can convert $ext{cos}^2 x$ using the identity $ext{cos}^2 x = 1 - ext{sin}^2 x$; 4. Substituting this provides:

ext{LHS} = ext{sin} x + 1 - ext{sin}^2 x - rac{1}{ ext{sin} x}; 5. This simplifies down to the right-hand side:

$= ext{cosec} x$.

Thus verified: