4.1 Vereenvoudig: cot² A + sin² A + cos² A · tan A 4.2 Bewys dat: \(rac{sin²(π + θ) + cos(180° - θ) - sec(360° - θ)}{tan(2π - θ) · cot(180° + θ)} = cos² θ\) - NSC Technical Mathematics - Question 4 - 2022 - Paper 2

To simplify the expression, we start with:

$ext{cot}^2 A + ext{sin}^2 A + ext{cos}^2 A an A$

Using the identity
$ext{cot} A = \frac{ ext{cos} A}{ ext{sin} A}, \text{and} \tan A = \frac{ ext{sin} A}{ ext{cos} A}$,

we can rewrite the expression as:

$\frac{ ext{cos}^2 A}{ ext{sin}^2 A} + ext{sin}^2 A + ext{cos}^2 A \cdot \frac{ ext{sin} A}{ ext{cos} A}$

Notice that: $\text{cos}^2 A · \frac{\text{sin} A}{\text{cos} A} = \text{cos} A \cdot \text{sin} A$

Thus, the expression can be rearranged to:

$\text{cot}^2 A + \text{sin}^2 A + \text{cos} A \cdot \text{sin} A$

Combining \text{sin}^2 A and \text{cos}^2 A using the Pythagorean identity: $\text{sin}^2 A + \text{cos}^2 A = 1$

So we simplify further:

$\text{cot}^2 A + 1$

Since (\text{cot}^2 A + 1 = \text{csc}^2 A), the final simplified form is:

$1$.