<s>[INST] How to prove expression don't relate to value of $k$ Given: $$\cfrac{1 - \cos (x) + k \sin (x) }{\sin (x) + k + k \cos (x) }$$
How to prove expression don't relate to value of $k$
Solve without using $\sin(2x) = 2 \sin(x)\cos(x)$
Please help.
 [/INST] $$\begin{align}\frac{1-\cos x+k\sin x}{\sin x+k(1+\cos x)}&=\frac{1-\cos x}{1-\cos x}\cdot \frac{1-\cos x+k\sin x}{\sin x+k(1+\cos x)}\\&=\frac{(1-\cos x)(1-\cos x+k\sin x)}{(1-\cos x)\sin x+k\sin^ 2x}\\&=\frac{(1-\cos x)(1-\cos x+k\sin x)}{\sin x(1-\cos x+k\sin x)}\\&=\frac{1-\cos x}{\sin x}\end{align}$$
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