If y= 1+cosx/sinx ,show that dy/dx=1/cosx-1

Question

If y= 1+cosx/sinx ,

show that dy/dx=1/cosx-1

Answer 1

Hello Fareem,

 

I am putting parentheses around certain terms.  This helps clarify the problem. In this case it seems to have helped.

 

y= (1+cos(x))/sin(x) = (1 + cos(x)) * (sin(x))^-1

 

Using the multiplication rule for derivatives:

 

dy/dx = -sin(x)/sin(x) - (1+cos(x))/sin²(x) * cos(x)

 

          = -1 - (cos(x) + cos²(x))/(1 - cos²(X))

 

          = -(1 - cos²(x) + cos(x) + cos²(X))/((1 + cos(x))(1 - cos(x)))

 

          = -(1 + cos(x))/((1 + cos(x))(1 - cos(x)))

 

          = -1/(1 - cos(x)) = 1/(cos(x) - 1)

Answer 2

Does y = (1+cos x)/sin x?  (Otherwise your function should have been simplified as y = 1 + cot x)

 

Just use the quotient rule, once you are sure of what your y expression is.

 

In the proposed dy/dx, is it supposed to be 1(cos x =1)? If so, parentheses are necessary to establish correct grouping.