I am interested in simplification for:
$$\frac{\sin\left(s\right)}{\cos\left(s\right)-1}$$
With Wolfram I know the correct answer is:
$$-\cot\left(\frac{s}{2}\right)$$
How to make this step by step?
Simplify $\frac{\sin\left(s\right)}{\cos\left(s\right)-1}$
trigonometry
Related Solutions
Proof without words: $\displaystyle\cot(\theta/2)=\frac{\color{green}{\sin(\theta)}}{\color{red}{1-\cos(\theta)}}$
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Suppose $\cos^{-1}\frac17\le x\le\frac\pi2$, and take $\theta$ in $[0,\frac\pi2]$ such that $\cos\theta=7\cos x$. Then $$\eqalign{ \frac{7}{2}(1+\cos 2 x)+{}&\sqrt{\left(\sin ^2 x-48 \cos ^2 x\right)} \sin x\cr &=(7 \cos x)(\cos x)+\sqrt{1-49 \cos ^2 x}\sin x\cr &=\cos x\cos\theta+\sin x\sin\theta\cr &=\cos(x-\theta)\cr}$$ and so $$\cos^{-1}({\rm this})=x-\theta=x-\cos^{-1}(7\cos x)\ ,$$ provided $0\le x-\cos^{-1}(7\cos x)\le\pi$, and this is true for all $x$ in the range stated above.
Best Answer
You can use the fact that$$\sin(s)=2\sin\left(\frac s2\right)\cos\left(\frac s2\right),$$whereas\begin{align}\cos(s)-1&=\cos^2\left(\frac s2\right)-\sin^2\left(\frac s2\right)-\left(\cos^2\left(\frac s2\right)+\sin^2\left(\frac s2\right)\right)\\&=-2\sin^2\left(\frac s2\right).\end{align}