Complete the following: 4.1.1 cosec A = .. - NSC Technical Mathematics - Question 4 - 2023 - Paper 2

Using the periodicity of the cosine function, we find:

$cos(2π + A) = cos A$

Using the property of cosecant:

$cosec(180° + A) = -cosec A$

First, apply trigonometric identities:

1. $sin(180° + A) = -sin A$
2. $cot(360° - A) = cot A$
3. $cos(2π - A) = cos A$

Now, substituting these values, we simplify to:

$-sin A \cdot cot A \cdot cos A + sin(3(360° - A))$

The sine function yields: $sin(3(360° - A)) = sin(-3A) = -sin(3A)$

Combining these, we have: $-sin A \cdot cot A \cdot cos A - sin(3A)$

Starting with the expression:

1. $cosec x - cosec x \cdot sec x = cosec x(1 - sec x)$

Using the identity for sec x: $sec x = \frac{1}{cos x}$

This gives: $tan² x + 1 = \frac{sin² x + cos² x}{cos² x} = 1/cos² x$ Thus confirming the identity