Question

Which of the following shows the simplified form of sine of x over the quantity 1 minus cosine of x end quantity?

Answer 1

- The solution steps are given below:

`\begin{gathered} (\sin x)/(1-\cos x) \\ \\ \text{ Multiply both numerator and denominator by }(1+\cos x) \\ \\ =(\sin x)/(1-\cos x)*((1+\cos x))/((1+\cos x)) \\ \\ =(\sin x+\cos x\sin x)/(1-\cos^2x) \\ \\ \text{ But we have:} \\ \sin^2x=1-\cos^2x\text{ \lparen From Trigonometric identities\rparen} \\ \\ =(\sin x+\cos x\sin x)/(\sin^2x) \\ \\ =(\sin x)/(\sin^2x)+(\cos x\sin x)/(\sin^2x) \\ \\ =(1)/(\sin x)+(\cos x)/(\sin x) \\ \\ =\csc x+\cot x \end{gathered}`

Final Answer

The answer is:

`\csc x+\cot x`