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How to one can show that the value of the following expression $$\frac{1-\cos(x)+k\sin(x)}{\sin(x)+k(1+\cos(x))}$$ doesn't depend to values of $k$?

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Writing $x = 2y$, we obtain

$$\begin{align} \frac{1-\cos x + k\sin x}{\sin x + k(1+\cos x)} &= \frac{(1-\cos (2y)) + k\sin (2y)}{\sin(2y) + k(1+\cos(2y))}\\ &= \frac{2\sin^2 y + 2k\sin y\cos y}{2\sin y\cos y + 2k\cos^2 y}\\ &= \frac{\sin y}{\cos y}\cdot\frac{\sin y+k\cos y}{\sin y + k\cos y}\\ &= \tan y \end{align}$$

using the double-angle formulae $\sin (2y) = 2\sin y\cos y$ and $\cos (2y) = \cos^2 y - \sin^2 y$.

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If $\frac{A}{B}=\frac{C}{D}$, for example $\frac{2}{3}=\frac{4}{6}$, then both fractions are also equal to $\frac{A+C}{B+D}$, or $\frac{6}{9}$ in my example.
Since $\frac{C}{D}=\frac{kC}{kD}$, it also equals $\frac{A+kC}{B+kD}$.
All you have to do is check that $A/B=C/D$.

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  • $\begingroup$ very neat solution! $\endgroup$ – Mufasa Sep 24 '13 at 12:04
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We have $\frac{1-\cos x + k \sin x}{\sin x + k(1+\cos x)}$ and what bothers me at first sight is that $k$ multiplies $\sin$ in numerator but $\cos$ in denominator. So, let's turn $\cos$ into $\sin$:

$$ \begin{align}\frac{1-\cos x + k \sin x}{\sin x + k(1+\cos x)} &= \frac{1-\cos x + k \sin x}{\sin x + k(1+\cos x)}\cdot \frac{1-\cos x}{1-\cos x}\\ &= \frac{(1-\cos x)(1-\cos x + k \sin x)}{(1-\cos x)\sin x + k(1-\cos^2 x)}\\ &= \frac{(1-\cos x)(1-\cos x + k \sin x)}{(1-\cos x)\sin x + k\sin^2 x}\\ &= \frac{(1-\cos x)(1-\cos x + k \sin x)}{\sin x(1-\cos x + k\sin x)}\\ &= \frac{1-\cos x}{\sin x} \end{align} $$

EDIT: Exactly the same technique can be used to solve Narasimham's problem as well.

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Transpose one term and split

$$ \sin^2 x + \cos^2 x = 1 $$

into two terms and place them in numerator (N) and denominator (D) to tally by cross multiplication

$$ \dfrac {1- \cos x }{\sin x } = \dfrac {\sin x }{1 + \cos x }= \dfrac {k \sin x }{k(1 + \cos x )} = \dfrac {1- \cos x + k \sin x }{\sin x + k(1 + \cos x )}=\tan \dfrac{x}2 $$

where we have multiplied both N and D of central fraction by a common factor $k$ and added up separately the N and D (which is a valid procedure with fractions) leaving its value unchanged for all values of $k$.

To learn further about this interesting property, I am tempted to say, it can also be seen equal to $ k^{th}$ root of:

$$ \dfrac {(1- \cos x)^k + \sin ^{k} x }{ \sin ^k x + { (1 + \cos x) }^k } = \tan \dfrac{x}2$$ for all values of $k$, a result burnt into my memory from Hall & Knight's Algebra text-book.

EDIT1:

As each fraction equals $\tan \dfrac{x}2$ it is included to show independence of $k$ directly.

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