Question

Differentiate y equals the quotient of the quantity 1 plus sine x and the quantity 1 minus cosine x.

–1

–2csc(x)sec(x)

2csc(x)sec(x)

the quotient of negative 1 times sine x plus cosine x minus 1 and the square of the quantity 1 minus cosine x

Answer 1

trig was never my strong point but ok


remember the quotient rule

`(dy)/(dx) (f(x))/(g(x))=(f'(x)g(x)-g'(x)f(x))/((g(x))^2)`

so
remember the pythagorean identity sin²(x)+cos²(x)=1
so


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

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

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

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

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

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

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

taht is the last option
thanks to jdoe0001 for showing me which identity to use