Answer: a) cot x - cot x cos x
Step-by-step explanation:
[tex]\dfrac{\sin x}{\sec x+1}\bigg(\dfrac{\sec x-1}{\sec x-1}\bigg)\\\\\\=\dfrac{\sin x(\sec x-1)}{\tan^2 x}\\\\\\=\dfrac{\sin x\sec x-\sin x}{\tan^2 x}\\\\\\=\dfrac{\sin x\sec x}{\tan^2}-\dfrac{\sin x}{\tan^2 x}\\\\\\=\dfrac{\frac{\sin x}{\cos x}}{\frac{\sin x}{\cos x}\tan x}-\dfrac{\sin x}{(\frac{\sin x}{\cos x})^2}\\\\\\=\dfrac{1}{\tan x}-\dfrac{\cos x\cos x\sin x}{\sin x\sin x}\\\\\\=\large\boxed{\cot x-\cot x\cos x}[/tex]
