Question

If Siny = x Sin (a+y). Prove that
dy/dx=[Sin²(a+y)]/Sina​

Answer 1

Use Trigonometric formulae to simplify expressions

Explanation:

Re-order to group the x and y,

Sin(y) / Sin(a+y) = x

to differentiate apply d/dx to both sides

`(d)/(dx)`Sin(y) / Sin(a+y) =
`(d)/(dx)`x

use division rule on left side which will give

`(dy)/(dx)`Cos(y)Sin(a+y) -
`(dy)/(dx)`Cos(a+y)Sin(y) / Sin²(a+y) = 1

Group them so dy/dx is on the left hand side:

`(dy)/(dx)` = Sin²(a+y) / Cos(y)Sin(a+y) - Cos(a+y)Sin(y)

Apply trigonometric sum formula to expand Sin(a+y) and Cos(a+y),

`(dy)/(dx)` = Sin²(a+y) / Cos(y)[Sin(a)Cos(y) + Cos(a)Sin(y)] - Sin(y)[Cos(a)Cos(y) - Sin(a)Sin(y)]

Simplify it

`(dy)/(dx)` = Sin²(a+y) / (Cos²(y) + Sin²(y))(Sin(a))

`(dy)/(dx)` = Sin²(a+y) / (1)(Sin(a))