Given that x = cot y, show that dy/dx = -1/(1+x^2)
- Identify that we are looking at dy/dx, not dx/dy and realise the relationship that dy/dx=1/(dx/dy)2)Try find dx/dy;cot = 1/tan or (tan)-1Hence, x=(tan y )-1 implying dx/dy = (-1)(tan y)-2(sec2 y ) =(-1)(sec2 y)/(tan2 y )Given 1+tan2=sec2, [from students memory or able to derive from cos2 + sin2 = 1] we get dx/dy=(-1)(1+tan2 y)/(tan2 y)and dy/dx= (-1)(tan2 y)/(1+tan2 y), dividing through by tan2 y, givesdy/dx = (-1)/(cot2 y + 1 ) and as x = cot y, dy/dx = -1/(1+x2) as required. 3) Alternatively as differential of cot is given as -cosec2 , we have;dx/dy= - cosec2(y) , hence dy/dx=(-1)/(cosec2(y)), and as cot2 (y)+1=cosec2(y)we get dy/dx=(-1)/(cot2 (y)+1), and so dy/dx = -1/(1+x2)
JF
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