Differentiate dy/dx 9 cos x sin y = 2

Question

I got the fraction form of the answer but the right option was like tan(x)tan(y).....not sure how tan comes how into play here in the end

Answer 1

This is an implicit differentiation problem requiring the use of the product rule:

9[cos(x)cos(y) y' + sin(y)(-sin(x)) = 0 ; used product rule on cosx sin y

cos(x)cos(y)y' = sin(x)sin(y) ; divided both sides by 9, then transposed terms containing sin to right

y' = sin(x)sin(y) / cos(x)cos(y) ; divided both sides by cosxcosy

y' = tan(x)tan(y) ; used trig identity tanu = sinu/cosu (twice)