Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$\frac{\tan x}{1-\cot x}+\frac{\cot x}{1-\tan x}=$	A. 1 + sec x . csc x
B. 1 + sin x . cos x
C. 1 + sin x . csc x
D. 1 + sec x . cos x
E. 1 − tan² x

$\begin{aligned}&\frac{\tan x}{1-\cot x}+\frac{\cot x}{1-\tan x}=\frac{\frac{\sin x}{\cos x}}{1-\frac{\cos x}{\sin x}}+\frac{\frac{\cos x}{\sin x}}{1-\frac{\sin x}{\cos x}}\\&=\frac{\frac{\sin x}{\cos x}}{\frac{\sin x-\cos x}{\sin x}}+\frac{\frac{\cos x}{\sin x}}{\frac{\cos x-\sin x}{\cos x}}=\frac{\sin^2x}{\cos x(\sin x-\cos x)}+\frac{\cos^2x}{\sin x(\cos x-\sin x)}\\&=\frac{\sin^3x-\cos^3x}{\sin x.\cos x(\sin x-\cos x)}\\&=\frac{(\sin x-\cos x)(\sin^2x+\sin x.\cos x+\cos^2x)}{\sin x.\cos x(\sin x-\cos x)}\\&=\frac{1+\sin x.\cos x}{\sin x.\cos x}=1+\sec x.\csc x\end{aligned}$