Which Of The Following Would Be An Acceptable First Step In Simplifying The Expression ?

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Question

Which of the following would be an acceptable first step in simplifying the expression sin x/ 1-cos x

Answer

the question does not present the options, but this does not interfere with the resolutionwe havesin x/ (1-cos x)we know thatsin²x+cos²x=1———–> sin²x=1-cos²xanddifference of squares(a+b)*(a-b)=a²-b²soThe idea is to make the difference of squares (1-cos²x)in denominator.somultiply the expression by (1+cos x)/(1+cos x)[sin x/ (1-cos x)]*[(1+cos x)/(1+cos x)]=[sin x*(1+cos x)]/[ (1-cos x)/(1+cos x)]=[sin x*(1+cos x)]/[ (1-cos²x)]=[sin x*(1+cos x)]/[ sin²x]=(1+cos x)/sin x=(1/sin x)+(cos x/sin x)=csc x+cot xthereforethe answer isthe first step is multiply the expression by (1+cos x)/(1+cos x)

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