Write the expression in terms of sine and cosine, then simplify : (sinx)/(cscx - cotx)

Question

write the expression in terms of sine and cosine, then simplify :
`(sinx)/(cscx - cotx)`

Answer 1

`1+\cos x`

Explanation:

`\boxed{\begin{minipage}{4cm}\underline{Trigonometric Identities}\\\\$\csc \theta=(1)/(\sin \theta)\\\\\\\cot \theta=(\cos \theta)/(\sin\theta)\\\\\\\cos^2 \theta + \sin^2 \theta = 1$\\\end{minipage}}`

`\begin{aligned}\implies ( \sin x)/( \csc x - \cot x) & = ( \sin x)/((1)/(\sin x) - (\cos x)/( \sin x))\\\\& = ( \sin x)/((1 - \cos x)/(\sin x))\\\\& = \sin x * (\sin x)/(1 - \cos x)\\\\& = (\sin^2 x)/(1-\cos x)\\\\& = (1-\cos^2x)/(1-\cos x)\\\\& = ((1-\cos x)(1+ \cos x))/(1-\cos x)\\\\& = 1 + \cos x\end{aligned}`