Question

1. given the identity: \\(\frac{1 + \tan x}{\sec x} = \sin x + \cos x\\)\

a) prove this identity. (2 marks)

Explanation:

Step1: Rewrite using sine/cosine

$\tan x = \frac{\sin x}{\cos x}$, $\sec x = \frac{1}{\cos x}$. Substitute into left side:
$\frac{1 + \frac{\sin x}{\cos x}}{\frac{1}{\cos x}}$

Step2: Simplify numerator

Combine terms in numerator:
$\frac{\frac{\cos x + \sin x}{\cos x}}{\frac{1}{\cos x}}$

Step3: Divide fractions

Multiply by reciprocal of denominator:
$\frac{\cos x + \sin x}{\cos x} \times \cos x = \sin x + \cos x$

Answer:

The identity is proven.