Question

Which of the following shows the simplified form of (sin x)/(1 - cos x)? 1sin x + tan xsin x + cot xcsc x + cot x

Answer 1

We want to simplify the following expression

`(\sin x)/(1-\cos x)`

We can start by multiplying both numerator and denominator by the conjugate of the denominator:

`\begin{gathered} (\sin(x))/(1-\cos(x))=(\sin(x))/(1-\cos(x))\cdot(1+\cos(x))/(1+\cos(x)) \\ \\ =(\sin(x)(1+\cos(x)))/((1-\cos(x))(1+\cos(x))) \\ \\ =(\sin(x)\cdot(1)+\sin(x)\cdot\cos(x))/((1)\cdot(1)+(1)\cdot(\cos(x))+(-\cos(x))\cdot(1)+(-\cos(x))\cdot(\cos(x))) \\ \\ \begin{equation*} =(\sin(x)+\sin(x)\cos(x))/(1-\cos^2(x)) \end{equation*} \end{gathered}`

Then, using the identity

`\sin^2x+\cos^2x=1\implies\sin^2x=1-\cos^2x`

We can rewrite our expression as

`(\sin(x)+\sin(x)\cos(x))/(1-\cos^2(x))=(\sin(x)+\sin(x)\cos(x))/(\sin^2(x))=(1)/(\sin x)+(\cos x)/(\sin x)`

By definition of cosecant and cotangent, our expression can be written as

`(1)/(\sin x)+(\cos x)/(\sin x)=\csc x+\cot x`

and this is our answer.

`(\sin x)/(1-\cos x)=\csc x+\cot x`